Differential Geometry For Physicists And Mathematicians Moving F

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1DIFFERENTIALGEOMETRYFORPHYSICISTSANDMATHEMATICIANSMovingFramesandDifferentialForms:FromEuclidPastRiemann8888_9789814566391_tp.indd124/12/1310:02am

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3JoséG.VargasPSTAssociates,LLC,USADIFFERENTIALGEOMETRYFORPHYSICISTSANDMATHEMATICIANSMovingFramesandDifferentialForms:FromEuclidPastRiemannWorldScientificNEWJERSEY•LONDON•SINGAPORE•BEIJING•SHANGHAI•HONGKONG•TAIPEI•CHENNAI8888_9789814566391_tp.indd224/12/1310:02am

4PublishedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HELibraryofCongressCataloging-in-PublicationDataVargas,JoséG.Differentialgeometryforphysicistsandmathematicians:movingframesanddifferentialforms:fromEuclidpastRiemann/byJoséGVargas(PSTAssociates,LLC,USA).pagescmIncludesbibliographicalreferencesandindex.ISBN978-9814566391(hardcover:alk.paper)1.Mathematicalphysics.2.Geometry,Differential.I.Title.QC20.V272014516.3'6--dc232013048730BritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.Copyright©2014byWorldScientificPublishingCo.Pte.Ltd.Allrightsreserved.Thisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthepublisher.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,USA.Inthiscasepermissiontophotocopyisnotrequiredfromthepublisher.PrintedinSingapore

5ToMs.GailBujakeforhercontributioninmakingthisworldabetterplacethroughhersupportofscienceandtechnologyv

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7AcknowledgementsItwouldbedifficulttoacknowledgeindetailthemanypersonswhohavecon-tributedtothejourneythatthewritingofthisbookhasrepresented.Ineachnewphase,newnameshadtobeadded.Notallcontributionswereessential,butallofthemimportant.Differentformsofinspiration,mathematicalorpedagogical,directorindi-rect,personalorthroughtheirwritings(inthecaseofthedeceased)areduetofamousmathematicianslikeE.CartanandE.K¨´ahler,butalsolessknownfig-ureslikethelateProfessorsY.H.Clifton(amathematicianatthedepartmentofPhysicsofUtahState)andFernandoSenent,physicistattheUniversityofValenciainSpain.MoralsupportandencouragementthroughtheyearsareduetoProfessorsDouglasG.TorrandAlwynvanderMerwe.ForavarietyofreasonssupportisalsoduetoDoctorsand/orProfessorsJafarAmirzadeh,VladimirBalan,HowardBrandt,IulianC.Bandac,ZbigniewOziewicz,MarshaTorrandYaohuanXu,andtoMs.LuminitaTeodorescu.PhaseSpaceTimeAssociatesprovidedgeneroussupport.Andlastbutmost,IacknowledgemywifeMayraforherpatienceandunderstanding.vii

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9PrefaceTheprinciplethatinformsthisbook.Thisisabookondifferentialgeometrythatusesthemethodofmovingframesandtheexteriorcalculusthroughout.Thatmaybecommontoafewworks.Whatisspecialaboutthisoneisthefollowing.Afterintroducingthebasictheoryofdifferentialformsandpertinentalgebra,westudythe“flatcases”knownasaffineandEuclideanspaces,andsimpleexamplesoftheirgeneralizations.Insodoing,weseekunderstandingofadvancedconceptsbyfirstdealingwiththeminsimplestructures.Differen-tialgeometrybooksoftenresorttoformaldefinitionsofbundles,Liealgebras,etc.thatarebestunderstoodbydiscoveringtheminanaturalwayincasesofinterest.Thosebooksthenprovideveryreconditeexamplesfortheillustra-tionofadvancedconcepts,saytorsion,eventhoughverysimpleexamplesexist.Misunderstandingsensue.In1492ChristopherColumbuscrossedtheAtlanticusinganaffineconnec-tioninasimplifiedform(aconnectionisnothingbutaruletonavigateamanifold).Heaskedthecaptainsoftheothertwoshipsinhissmallflotillatoalwaysmaintainwhatheconsideredtobethesamedirection:West.Thatconnectionhastorsion.ElieCartanintroduceditinthemathematicalliterature´centurieslater[13].Wecanlearnconnectionsfromapracticalpointofview,thepracticaloneofColumbus.Thatwillhelpustoeasilyunderstandconceptslikeframebundle,connection,valuedness,Liealgebra,etc.,whichmightoth-erwiselookintimidating.Thus,forexample,weshallslowlyacquireagoodunderstandingofaffineconnectionsasdifferential1−formsintheaffineframebundleofadifferentiablemanifoldtakingvaluesintheLiealgebraoftheaffinegroupandhavingsuchandsuchproperties.ReplacethetermaffinewiththetermsEuclidean,conformal,projective,etc.andyouhaveenteredthetheoriesofEuclidean,projective,conformal...connections.Cartan’sversusthemodernapproachtogeometry.ItissometimesstatedthatE.Cartan’sworkwasnotrigorous,andthatitisnotpossibletomakeit´so.Thisstatementhasledtothedevelopmentofothermethodstododiffer-entialgeometry,fullofdefinitionsanddistractingconcepts;notthestylethatphysicistslike.YeatonH.Cliftonwasagreatdifferentialtopologist,anopinionofthisauthorwhichwasalsosharedbythewellknownlatemathematicianS.-S.Cherninprivateconversationwiththisauthor.CliftonhadoncetoldmethattheonlythingthatwasneededtomakerigorousCartan’stheoryofconnectionswasix

10xPREFACEtoaddacoupleofdefinitions.Afewyearslater,uponthepresentauthor’sprodding,Cliftondeliveredonhisclaim.Tobeprecise,heshowedthatjustamajordefinitionandacoupleoftheoremswereneeded.Theproofisinthepudding.Itisservedinthelastsectionofchapter8andinthesecondsectionofchapter9.Unfortunately,Cartan’sapproachhasvirtuallyvanishedfromthemodernliterature.AlmostacenturyafterhisformulationofthetheoryofaffineandEuclideanconnectionsasageneralizationofthegeometryofaffineandEu-clideanspaces[11],[12],[14],anupdateisdueonhisstrategyforthestudyofgeneralizedspaceswiththemethodofthemovingframe[20].Weshallfirststudyfromtheperspectiveofbundlesandintegrabilityofequationstwoflatgeometries(theirtechnicalnameisKleingeometries)andthenproceedwiththeirCartangeneralization.InthoseKleingeometries,affineandEuclidean,conceptslikeequationsofstructurealreadyexist,andthemathematicalexpres-sionofconceptslikecurvatureandtorsionalreadyariseinfull-fledgedform.Itsimplyhappensthattheytakenullvalues.Mathematicalsubstanceunderlyingthenotation.Thereisaprofounddif-ferencebetweenmostmodernpresentationsandours.Mostauthorstrytofiteverythingthattransformstensoriallyintothemoldof(p,q)−tensors(ptimescontravariantandqtimescovariant).FollowingK¨ahlerinhisgeneralizationofCartan’scalculus,[46],[47],[48],wedonotfindthattobetherightcourseofaction.Hereiswhy.Facedwithcovarianttensorfieldsthataretotallyskew-symmetric,themod-ernapproachthatwecriticizeignoresthatthenaturalderivativeofatensorfield,whetherskew-symmetricornot,isthecovariantderivative.Theyresorttoex-teriorderivatives,whichbelongtoexterioralgebra.Thatisunnaturalandonlycreatesconfusion.Exteriordifferentiationshouldbeappliedonlytoexteriordifferentialforms,andthesearenotskew-symmetrictensors.Theyonlylookthatway.Covarianttensorfieldshavesubscripts,butsodoexteriordifferentialforms.Formostoftheauthorsthatwecriticize,thecomponentsofthosetwotypesofmathematicalobjectshavesubscripts,whichtheycallqindices.Butnotalltheqindicesarebornequal.Therewillbeskew-symmetryandexteriordifferentiationinconnectionwithsomeofthem—“differentialform”subscripts—butnotinconnectionwiththeremainingones,whethertheyareskew-symmetricwithrespecttothoseindicesornot.Theyaretensorsubscripts.Likesuperscripts,theyareassociatedwithcovariantdifferentiation.Correspondingly,thecomponentsofquantitiesintheCartanandK¨ahlercalculushave—inadditiontoaseriesofsuperscripts—twoseriesofsubscripts,oneforintegrandsandanotheroneformultilinearfunctionsofvectors.ThisisexplicitlyexhibitedinK¨ahler[46],[47],[48].Theparagonofquantitieswiththreetypesofindices.Affinecurvatureisa(1,1)−tensor-valueddifferential2−form.Thefirst“1”inthepairisforasuperscript,andtheotheroneisforasubscript.Torsionsare(1,0)−valueddifferential2−formsandcontorsionsare(1,1)−valueddifferential1−forms.

11PREFACExiLetvrepresentvectorfieldsandletdbetheoperatorthatCartancallsexteriordifferentiation.dvisavector-valueddifferential1−form,andddvisavector-valueddifferential2−form.ExpertsnotusedtoCartan’snotationneedbeinformedthatddvis(vμRν)ωλ1∧ωλ2e.Relativetobasesof(p=1,μλ1λ2νq=0)−valueddifferential2−forms,thecomponentsofddvare(vμRν).μλ1λ2Onecanthendefinea(1,1)−valueddifferential2−formwhosecomponentsaretheRν’s,andwhoseevaluationonv(respondingtotheq=1partoftheμλ1λ2valuedness)yieldsddv.Hence,thetraditional(p,q)−characterizationfallsshortoftheneedforagoodunderstandingofissuesconcernedwiththecurvaturedifferentialform.Bundlesareoftheessence.Theperspectiveofvaluednessthatwehavejustmentionedisonewhichbestfitssectionsofframebundles,andtransformationsrelatingthosesections.Lestbeforgotten,thesetofallinertialframes(theydonotneedtobeinertial,butthatisthewayinwhichtheyappearinthephysicsliterature)constitutesaframebundle.Grosslyspeaking,abundleisasetwhoseelementsareorganizedlikethoseinertialframesare.Theonesatanygivenpointconstitutethefiberatthatpoint.Wehaveidenticalfibersatdifferentpoints.Theremustbeagroupactinginthebundle(likePoincar´e’sisinourexample),andasubgroupactinginthefibers(thehomogenousLorentzgroupinourexample).Aninterestingexampleofsectionofabundleisfoundincosmology.Oneiscomputinginaparticularsectionwhenonerefersquantitiestotheframeofreferenceofmatteratrestinthelarge.Asectionisbuiltwithoneandonlyoneframefromeachfiber,thechoicetakingplaceinacontinuousway.But,forfoundationalpurposes,itisbettertothinkintermsofthebundlethanofthesections.Atanadvancedlevel,onespeaksofLiealgebravaluednessofconnections,theLiealgebrabeingavectorspaceofthesamedimensionasthebundle.AllthisismuchsimplerthanitsoundswhenonereallyunderstandsEuclideanspace.Wewill.ItisunfortunatethatbooksonthegeometryofphysicsdealwithconnectionsvaluedinLiealgebraspertainingtoauxiliarybundles(i.e.notdirectlyrelatedtothetangentvectors)anddonotevenbotherwiththeLiealgebrasofbundlesofframesoftangentvectors.WhichphysicistevermentionswhatistheLiealgebrawheretheLevi-Civitaconnectiontakesitsvalues?Incidentally,thetangentvectorsthemselvesconstituteasocalledfiberbundle,eachfiberbeingconstitutedbyalltangentvectorsatanygivenpoint.Itisthetangentbundle.ThisauthorclaimsthatthegeometryofgroupssuchasSU(3)andU(1)×SU(2)fitsinappropriatelyextendedtangentbundlegeometry,ifonejustknowswheretolook.Onedoesnotneedauxiliarybundles.Thatwillnotbedealtwithinthisbook,butincomingpapers.ThisbookwilltellyouwhetherIdeserveyourtrustandshouldkeepfollowingmewhereIthinkthattheideasofEinstein,CartanandK¨ahlertakeus.AssumetherewereaviableoptionofrelatingU(1)×SU(2)×SU(3)tobundlesoftangentvectors,theirframes,etc.Itwouldbeunreasonabletoremainsatisfiedwithauxiliarybundles(Yang-Millstheory).Inanycase,oneshouldunderstand“mainbundlesgeometry”(i.e.directlyrelatedtothetangent

12xiiPREFACEbundle)beforestudyingandpassingjudgementonthemeritsanddangersofYang-Millstheory.Specificfeaturesdistinguishingthisbookareasfollows:1.Differentialgeometryispresentedfromtheperspectiveofintegrability,usingsocalledmovingframesinframebundles.ThesystemsofdifferentialequationsinquestionemergeinthestudyofaffineandEuclideanKleingeome-tries,thosespecificsystemsbeingintegrable.2.Inthisbook,itdoesnotsufficewhethertheequationsofthegeneralcase(curved)havetheappropriateflatlimit.Itisamatterofwhetherweuseinthegeneralcaseconceptswhicharethesameorascloseaspossibletotheintuitiveconceptsusedinflatgeometry.Thus,theall-pervasivedefinitionoftangentvectorsasdifferentialoperatorsinthemodernliteratureisinimicaltoourtreatment.3.Inthesamespiritoffacilitatingunderstandingbynon-mathematicians,differentialformsareviewedasfunctionsofcurves,surfacesandhypersurfaces[65](Weshallusethetermhypersurfacetorefertomanifoldsofarbitrarydimen-sionthatarenotKleinspaces).Inotherwords,theyarenotskew-symmetricmultilinearfunctionsofvectorsbutcochains.Thisbookcoversalmostthesamematerialasapreviousbookbythisauthor[85]exceptforthefollowing:1.Thecontentsofchapters1,3and12hasbeenchangedorextendedverysignificantly.2.Wehaveaddedtheappendices.AppendixApresentstheclassicaltheoryofcurvesandsurfaces,buttreatedinatotallynovelwaythroughtheintro-ductionoftheconceptofcanonicalframefieldofasurface(embeddedin3-DEuclideanspace).Wecouldhavemadeitintoonemorechapter,butwehavenotsinceconnectionsconnecttangentvectorsinthebookexceptinthatappendix;vectorsin3-DEuclideanspacethatarenottangentvectorstothespecificcurvesandsurfacesbeingconsideredareneverthelesspartofthesubjectmatter.AppendixBspeaksoftheworkofthemathematicalgeniusesElieCartan´andHermannGrassmann,inordertohonortheenormouspresenceoftheirideasinthisbook.AppendixCisthelistofpublicationsofthisauthorforthosewhowanttodealfurtherintotopicsnotfullyaddressedinthisbookbutdirectlyrelatedtoit.YoucanfindtherepapersonFinslergeometry,unificationwithteleparallelism,theK¨ahlercalculus,alternativestothebundleoforthonormalframes,etc.3.Severalsectionshavebeenaddedattheendofseveralchapters,touchingsubjectssuchasdiagonalizationofmetricsandorthonormalizationofframes,CliffordandLiealgebras,etc.

13ContentsDedicationvAcknowledgementsviiPrefaceixIINTRODUCTION11ORIENTATIONS31.1Selectivecapitalizationofsectiontitles...............31.2Classicalinclassicaldifferentialgeometry.............41.3Intendedreadersofthisbook....................51.4ThefoundationsofphysicsinthisBOOK.............61.5MathematicalVIRUSES.......................91.6FREQUENTMISCONCEPTIONS.................111.7Prerequisite,anticipatedmathematicalCONCEPTS.......14IITOOLS192DIFFERENTIALFORMS212.1Acquaintancewithdifferentialforms................212.2Differentiablemanifolds,pedestrianly................232.3Differential1−forms.........................252.4Differentialr−forms.........................302.5Exteriorproductsofdifferentialforms...............342.6Changeofbasisofdifferentialforms................352.7Differentialformsandmeasurement................372.8DifferentiablemanifoldsDEFINED.................382.9AnotherdefinitionofdifferentiableMANIFOLD..........40xiii

14xivCONTENTS3VECTORSPACESANDTENSORPRODUCTS433.1INTRODUCTION..........................433.2Vectorspaces(overthereals)....................453.3Dualvectorspaces..........................473.4Euclideanvectorspaces.......................483.4.1Definition...........................483.4.2Orthonormalbases......................493.4.3Reciprocalbases.......................503.4.4Orthogonalization......................523.5NotquiterightconceptofVECTORFIELD............553.6Tensorproducts:theoreticalminimum...............573.7FormalapproachtoTENSORS...................583.7.1Definitionoftensorspace..................583.7.2Transformationofcomponentsoftensors.........593.8Cliffordalgebra............................613.8.1Introduction.........................613.8.2BasicCliffordalgebra....................623.8.3ThetangentCliffordalgebraof3-DEuclideanvectorspace643.8.4ThetangentCliffordalgebraofspacetime.........653.8.5Concludingremarks.....................664EXTERIORDIFFERENTIATION674.1Introduction..............................674.2Disguisedexteriorderivative.....................674.3Theexteriorderivative........................694.4Coordinateindependentdefinitionofexteriorderivative.....704.5Stokestheorem............................714.6Differentialoperatorsinlanguageofforms.............734.7Theconservationlawforscalar-valuedness.............774.8LieGroupsandtheirLiealgebras..................79IIITWOKLEINGEOMETRIES835AFFINEKLEINGEOMETRY855.1AffineSpace..............................855.2Theframebundleofaffinespace..................875.3Thestructureofaffinespace.....................895.4Curvilinearcoordinates:holonomicbases.............915.5Generalvectorbasisfields......................955.6StructureofaffinespaceonSECTIONS..............975.7Differentialgeometryascalculus..................995.8InvarianceofconnectiondifferentialFORMS...........1015.9TheLiealgebraoftheaffinegroup.................1035.10TheMaurer-Cartanequations....................1055.11HORIZONTALDIFFERENTIALFORMS............107

15CONTENTSxv6EUCLIDEANKLEINGEOMETRY1096.1Euclideanspaceanditsframebundle................1096.2ExtensionofEuclideanbundletoaffinebundle..........1126.3Meaningsofcovariance........................1146.4Hodgedualityandstaroperator..................1166.5TheLaplacian.............................1196.6Euclideanstructureandintegrability................1216.7TheLiealgebraoftheEuclideangroup..............1236.8Scalar-valuedclifforms:K¨ahlercalculus..............1246.9Relationbetweenalgebraandgeometry..............125IVCARTANCONNECTIONS1277GENERALIZEDGEOMETRYMADESIMPLE1297.1Ofconnectionsandtopology....................1297.2Planes.................................1307.2.1TheEuclidean2-plane....................1317.2.2Post-Klein2-planewithEuclideanmetric.........1327.3The2-sphere.............................1347.3.1TheColumbusconnectiononthepunctured2-sphere...1347.3.2TheLevi-Civitaconnectiononthe2-sphere........1367.3.3Comparisonofconnectionsonthe2-sphere........1377.4The2-torus..............................1387.4.1Canonicalconnectionofthe2-torus............1387.4.2Canonicalconnectionofthemetricofthe2-torus.....1407.5AbridgedRiemann’sequivalenceproblem.............1407.6UseandmisuseofLevi-Civita....................1418AFFINECONNECTIONS1438.1Liedifferentiation,INVARIANTSandvectorfields........1438.2Affineconnectionsandequationsofstructure...........1478.3Tensorialityissuesandseconddifferentiations...........1508.4Developmentsandannulmentofconnection............1538.5Interpretationoftheaffinecurvature................1548.6Thecurvaturetensorfield......................1568.7Autoparallels.............................1588.8Bianchiidentities...........................1598.9Integrabilityandinterpretationofthetorsion...........1608.10Tensor-valuednessandtheconservationlaw............1618.11Thezero-torsioncase.........................1648.12Horriblecovariantderivatives....................1658.13Affineconnections:rigorousAPPROACH.............167

16xviCONTENTS9EUCLIDEANCONNECTIONS1719.1MetricsandtheEuclideanenvironment..............1719.2EuclideanstructureandBianchiIDENTITIES..........1739.3ThetwopiecesofaEuclideanconnection.............1779.4AffineextensionoftheLevi-Civitaconnection...........1789.5Computationofthecontorsion...................1799.6Levi-Civitaconnectionbyinspection................1809.7StationarycurvesandEuclideanAUTOPARALLELS......1859.8EuclideanandRiemanniancurvatures...............18810RIEMANNIANSPACESANDPSEUDO-SPACES19110.1KleingeometriesingreaterDETAIL................19110.2ThefalsespacesofRiemann.....................19310.3MethodofEQUIVALENCE.....................19510.4Riemannianspaces..........................19710.5Annulmentofconnectionatapoint................19910.6EmergenceandconservationofEinstein’stensor.........20110.7EINSTEIN’SDIFFERENTIAL3-FORM.............20210.8Einstein’s3−form:propertiesandequations............20510.9EinsteinequationsforSchwarzschild................208VTHEFUTURE?21311EXTENSIONSOFCARTAN21511.1INTRODUCTION..........................21511.2Cartan-Finsler-CLIFTON......................21611.3Cartan-KALUZA-KLEIN......................21811.4Cartan-Clifford-KAHLER......................220¨11.5Cartan-K¨ahler-Einstein-YANG-MILLS...............22112UNDERSTANDTHEPASTTOIMAGINETHEFUTURE22512.1Introduction..............................22512.2Historyofsomegeometry-relatedalgebra.............22512.3Historyofmoderncalculusanddifferentialforms.........22712.4HistoryofstandarddifferentialGEOMETRY...........22912.5Emergingunificationofcalculusandgeometry..........23312.6Imaginingthefuture.........................23513ABOOKOFFAREWELLS23713.1Introduction..............................23713.2Farewelltovectoralgebraandcalculus...............23713.3FarewelltocalculusofcomplexVARIABLE............23913.4FarewelltoDirac’sCALCULUS...................24013.5Farewelltotensorcalculus......................24213.6FarewelltoauxiliaryBUNDLES?..................243

17CONTENTSxviiAPPENDIXA:GEOMETRYOFCURVESANDSURFACES247A.1Introduction..............................247A.2Surfacesin3-DEuclideanspace...................248A.2.1Representationsofsurfaces;metrics............248A.2.2Normaltoasurface,orthonormalframes,area......250A.2.3TheequationsofGaussandWeingarten..........251A.3Curvesin3-DEuclideanspace...................252A.3.1Frenet’sframefieldandformulas..............252A.3.2Geodesicframefieldsandformulas.............253A.4Curvesonsurfacesin3-DEuclideanspace.............254A.4.1Canonicalframefieldofasurface..............254A.4.2Principalandtotalcurvatures;umbilics..........255A.4.3Euler’s,Meusnier’sandRodrigues’estheorems......256A.4.4Levi-Civitaconnectioninducedfrom3-DEuclideanspace256A.4.5TheoremaegregiumandCodazziequations........257A.4.6TheGauss-Bonnetformula.................257A.4.7Computationofthe“extrinsicconnection”ofasurface..259APPENDIXB:“BIOGRAPHIES”(“PUBLI”GRAPHIES)261B.1ElieJosephCartan(1869–1951)...................261B.1.1Introduction.........................261B.1.2Algebra............................262B.1.3Exteriordifferentialsystems.................263B.1.4Geniusevenifweignorehisworkingonalgebra,exteriorsystemsproperanddifferentialgeometry.........263B.1.5Differentialgeometry.....................264B.1.6Cartanthephysicist.....................265B.1.7Cartanascriticandmathematicaltechnician.......266B.1.8Cartanasawriter......................267B.1.9Summary...........................268B.2HermannGrassmann(1808–1877).................269B.2.1Minibiography........................269B.2.2Multiplicationsgalore....................269B.2.3Tensorandquotientalgebras................270B.2.4Impactandhistoricalcontext................271APPENDIXC:PUBLICATIONSBYTHEAUTHOR273References277Index285

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19PartIINTRODUCTION1

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21Chapter1ORIENTATIONS1.1SelectivecapitalizationofsectiontitlesWhilethisbookisprimarilyintendedfornon-expertsondifferentialgeometry,expertsmayenjoythenumerousadvancedcommentssprinkledthroughthetext,whichonedoesnotfindtogetherinasinglesource.Somecommentsqualifyasadvancedsimplybecausetheyareabsentinotherapproachestodifferentialgeometry,butareessentialhere.Ifthosecommentswereomitted,thisbookwouldnothavebeenworthwriting.Theirabsencefrommostoftheliteraturemaybeattherootofnumerouserrorsfoundinit.Inordertoaccomodatethewideaudienceextendingfromneophytestoex-perts,wehaveincludedsectionsmeanttoachievegreaterrigorinsomekeyarea,buttooabstractfornon-experts.Wheneversuitable,thosesectionshavebeenplacedattheendofchapters.Wehaveusedselectivecapitalizationintitlesofsectionsasguidancetodegreeofdifficulty,thoughnouniversalagreementmaybeachievableinthisregard.Thus,wehavewritteninfullcapitalsthetitlesofsectionsthatshouldbeskippedbynon-expertreaders.Ifwethinkthatsomecommentsbutnotawholesectionaretooadvanced,wealertofthatfactbywritingthelastword(s)ofthetitleofthesectioninfullcapitals.Insection8.1,whichisanexception,themiddleratherthattheendpartshouldbeskipped,ifany.Thisbookshouldbereadlinearlyforthemostpart.However,linearunder-standingmaynotbeanachievablegoalformanyreaders.Thatisnoproblem.Mostofusdonotunderstandeverythingthefirsttimethatweencountersomenewtheory.Wekeeplearningthesamesubjectatevergreaterdepthsinsuc-cessiveapproximationstoacompleteunderstanding.Weadvocatethatnon-expertsskipthesectionswithtitlesinfullcapitals,andthat,whenreadingtheoneswhereonlypartofthetitleisincapitals,theydisregardanypassagethattheyfindobscure.Theyshouldtrytopayattentiontothebottomlineofarguments.Suchaskippingrisksthatareadermayoverlooksomethingimportant.Inordertominimizethisproblem,concepts3

224CHAPTER1.ORIENTATIONSandremarkshaveoftenbeenrepeatedthroughoutthetext.Learningdifferentialgeometryshouldnotbedifferentfromlearningaforeignlanguage.Afterlearningdifferentialgeometryatthelevelofthecommandofhismotherlanguagebyafiveyearold,onecanlearnthegrammarofdifferentialgeometrybyre-readingthisbookmorethoroughly.Thosewhoneedamorethoroughormoreprecisevocabulary—noteveryphysicistdoes—canalwaysreadotherbooksonthesubject.Thisoneisnotabible,theirvaluebeinginanycasereligiondependent.1.2Themeaningofclassicalin“classicaldifferentialgeometry”Onemightthinkthatthetermclassicaldifferentialgeometryreferstoallthosegeometriesthatpreceded(inphysics)theformulationofYang-Millsgeometry.ItcertainlymustincludethetheoryofcurvesandsurfacesinEuclideanspace,whichwasvirtuallycompletedbythemidnineteenthcentury(AppendixAdealswiththattheory,exceptinlanguagenotknownatthetime).OnewouldexpecttheitalicizedtermtoalsoincludetheRiemanniangeneralizationofEuclideangeometry,anditsfurthergeneralizationbyCartan.Butthatisnothowthetermisused.InRiemann’sandCartan’stheories,asurfaceisadifferentiablemanifoldofdimensiontwo,meaningthatweneedtwoindependentcoordinatestolabelitspoints.But,asweshallseeinanappendix,thetheoryofdifferentiablemanifoldsofdimensiontwoandthetheoryofsurfacesdevelopedbeforeRiemanndonotcoincide.Thus,whereaswemayspeakofthetorsionofadifferentiable2-manifold,wemaynotspeakofthetorsionofasurfaceinthetheoryofcurvesandsurfacesofthattime.Letuslookatthisissuemoreindetail.Ingeneraldifferentiablemanifolds,thereisnoroomforvectors.Forinstance,theydonotfitonthesphere(SpanishreadersnottooconversewiththeEnglishlanguageshouldnotethatwhatwecallesferatranslatesintoballinEnglish,andwhatwecallsphericalsurfacetranslatesintosphere).Vectorsstickoutofthesphereorarelikecords;iftheyareverysmall,wedonotnoticethis,whichisthereasonwhyitwasthoughtforagesthattheearthwasflat.Wemayanddo,however,speakoftangentvectorspreciselybecauseofthispropertyofsmallvectorslyingonthesurfaceasifitwereflat.Ofcourse,thisisnotarigorouswayofsayingthings;wewillbecomemorerigorouslateron.Aflagpole,lyingdownonaflatfloor,closelyrepresentstheconceptoftangentvectorforthepurposesofthissection.Aconnectionislikearuletocompareflagpoleslyingdownatdifferentpointsofasurface.Differentialgeometryinthemodernsenseofthewordismainlythestudyofsuchrules.Inthattheoryandinordertoavoidconfusionwithclassicaldifferentialgeometry.Weshouldthenspeakof2-dimensionalmanifoldsratherthansurfaces,atleastuntilweunderstandwhatisstudiedineachcase.Intheclassicaldifferentialtheoryofcurvesandsurfaces(i.e.geometry),we

231.3.INTENDEDREADERSOFTHISBOOK5carealsoaboutrulesforuprightpoles,orstickingatanangle.Thoserulescannotbeassignedspecificallytothesurface,buttothelatterbeinglivingin3-DEuclideanspace.Somethingsimilaroccurswithcurves,whichareone-dimensionalmanifolds.Whenonereferstothe(ingeneral,notnull)curvatureandtorsionofcurves,oneisreferringtoconceptsdifferentfromwhattheyareindifferentiablemanifoldtheory.Thedifferencesjuststatedbetweenthetheoryofcurvesandsurfaces,ontheonehand,andthetheoryofdifferentiablemanifoldsofdimensions1and2,ontheotherhand,hasledustorelegatethefirstofthosetoAppendixA.Havingmadetheforegoingdistinctions,weshallusethetermsurfaceinbothcontexts,i.e.whetherinclassicaldifferentialgeometryorinmodernone.Considernowthegeometryassociatedwiththeelectroweakandstrongin-teractions.Herewearealsodealingwithmoderntheoryofconnectionsondifferentiablemanifolds,butoneisnotconnectingtangentvectorsbutvectorsofsomeauxiliaryspaces,andframesofauxiliarybundlesconstructeduponthosespaces.Auxiliarymeanspreciselynotbeingdirectlyrelatedtotangentvectors.Thisbookisaboutmoderndifferentialgeometryasconceivedand,equallyimportant,aspresentedbythegreatmasterElieCartan(hisson,HenriCartan,´alsowasafamousmathematician,cofounderoftheBourbakigroup;inthisbook,wealwaysrefertoElie,notHenri).Themainconceptisthe´connectiononabundleofframesmadeofvectorstangenttoadifferentiablemanifold.ReadersreachingallthewaytotheappendicesshouldbewellpreparedtofollowtheargumentthatonedoesnotneedtoinvokeauxiliarybundlestodowhatYang-Millstheorypresentlydoes.Bythetimethatthisbookgoestopressandfallsinyourhands,somepapermakingthatpointshouldhavebeenpostedbythisauthorinthearXiv,orprintedsomewhereelse.1.3IntendedreadersofthisbookThebackboneofthisbookisthetheoryofaffineandEuclideanconnections.Thereareotherbooksonthesubject,buttheytargetareadershipinthemodernstyleofdoingmathematics.Weintendtoservereadersunsatisfiedbythatstyle.Contrarytomostmodernapproaches,weusethesameconceptsofvectorfieldandconnectionasinthefirstcoursesofcalculus,i.e.asinthestudyof3-DEuclideanspace.Inthoseelementarytreatments,vectorfieldsarenotdiffer-entiableoperators,norwilltheybesohere,butpassiveobjectsnotactingonanything.Theyareactuallyactedupon,differentiated,tobespecific.Connec-tionsarenotexplicitlymentionedinthosetreatments;theyarenotanabsentconcept,buttheirsimplicityandthelackofthoroughnessofstandardtreat-mentsmakesthembeoverlooked.HereconnectionsaremadeexpliciteveninaffineandEuclideanspace.Thesignaturecalculusofdifferentialgeometryistheexteriorcalculus.Itdealswithexpressionssuchasf(x,y,z)dx+g(x,y,z)dy,ordx∧dy,etc.,towhichweshallreferasscalarvalueddifferentialformsorsimplydifferentialforms.Unlikedxi+dyj+dzk,whichweshallcallvectorvalued,theycontainno

246CHAPTER1.ORIENTATIONSvectorsorvectorfields.Inourapproach,differentialgeometryis,infirstap-proximation,thecalculusofvector-valueddifferentialforms,whichincludestheexteriorcalculusofscalar-valuedones.Inordertodifferentiatevectorfieldsorvector-valueddifferentialforms,oneneedstoconnectvectorsatnearbypoints.Aconnectionisthusinvolved.Othertypesofvaluednessalsoareconsidered:covariant,mixedcontravariant-covariant,Lie-algebravaluedness,Cliffordalge-bravaluedness,etc.Theywillemergeinanorganicwaystartingwiththedifferentiationofcontravariant(readordinary)vectorfields..Wedealinextensowiththeconceptsofdifferentialformsanddifferentiablemanifoldsinchapter2.Theexteriorcalculusisintroducedinchapter4.Mostofthealgebraunderlyingthiscalculusisgiveninchapter3.AffineandEuclideanspacesarerespectivelytreatedinchapters5and6.Fortheuninitiated,affinespaceisjustEuclideanspacewiththeconceptofdistanceremovedfromit.Theirgeneralizationsarethedifferentiablemanifoldsendowedwithaffineconnections(chapter8)andEuclideanconnections(chapters9and10).Chapter7isatransitionalchapterconstitutedbysimpleexamplesofconnecteddifferentiablemanifolds.Fromtheprecedingconsiderationsfollowsthatourintendedreadersaremainlyadvancedphysicsandmathematicsstudents,butalsostudentsofothersciencesandengineeringwithunusualmathematicsorphysicsinterest.Becauseofthetechniqueweuse,theyshouldbematureenoughnottobediscouragedbyunfamiliaritywiththiscalculus.Onceagain—foravarietyofreasons(andfol-lowingRudin’sbabybookonanalysis[65]andthewaysofCartanandK¨ahler)—ourdifferentialformsareintegrands,i.e.functionsoflines,surfaces,volumes,etc.(cochainsinthemainliterature),notantisymmetricmultilinearfunctionsofvectors.Althoughsuchadvancedstudentsconstitutethemainaudiencewetarget,wehopethatprofessionalsmayfindthisapproachtoberefreshingbecauseoftheabundanceofunusualadvancedconcepts.Finally,chapters11-13,speciallythelastone,providesaglimpseofthemanydoorsthatthisbookopens.1.4ThefoundationsofphysicsinthisBOOKThissectionandthenexttworeflectthefactthatthisbookandothertofollowarewhattheyarebecausethisauthorfoundthatthemathematicsalreadyexiststocarryoutwhatEinsteincalledthelogicalhomogeneityofdifferentialgeometryandthefoundationsofphysics(Einsteinusedthetermtheoreticalphysics).Inmodernlanguage,thatwouldbetheidentificationoftheequationsofstructureofsomegeometrywiththefieldequationsofthephysics.Judgingthesources(moreaccurately,lackthereof)usedtotrainphysicistsinthefoundationsofphysics,onecanunderstandwhythatidentificationmaybeconsideredmissionimpossibleabinitio.Weshallpresentacriticalpanoramicviewoftheoreticalandfoundationsphysics(thissection)andpointoutimped-imentstoitsprogress(nexttwosections).Maythisbookstarttopersuadeitsreadersthat,afterall,Einstein’sproposalforunificationfromteleparallelism

251.4.THEFOUNDATIONSOFPHYSICSINTHISBOOK7mightbetherightwaytogo.Inthelastdecadesphysicshasnotsolvedonesingle“theoryindependent”problem,likethemassoftheelectronis.Norhasitsolved“structuralprob-lems”ofphysicaltheory,likethoseweareabouttomention.Muchheraldednewtheoriesthatpromisedtogiveuseverythingappeartobefizzlingaway.Somereaderswillcontextmynegativeperception.Butthereareauthoritativeopinionsthatareevenmorederogatorythanmine.Thus,forinstance,theveryeminentphysicistCarverMead(emeritusprofessoratCaltech,winnerofLemelson-MIT1999Prizeforinventionandinnovation,IEEEJohnvonNeu-mannMedal,founderofseveraltechnologycompanies)openstheintroductionofhiswonderfulbookCollectiveElectrodynamics[55]withthefollowingstate-ment:“Itismyfirmbeliefthatthelastsevendecadesofthetwentiethcenturywillbecharacterizedinhistoryasthedarkagesoftheoreticalphysics”.Thatwasin2000.Thereisnoreasonwhyhewouldnotbewritingtodayofmorethaneightdecadesofdarkages.IwouldqualifyMead’sstatementbydirectinghiscriticismtothefoundationsofphysics,nottotheoreticalphysics.Afterall,therehasbeenundeniableprogressinthefieldwhichgoesfarbeyondwhatissimplyexperimental,andforwhichwecannotthinkofanameifnottheoreticalphysics,evenifheavilyloadedwithphenomenology.Isubmitthattheexplosioninthedevelopmentoftechnologyoverthoseeightdecadeshasallowedtheoreticalphysicstomakeprogress—inspiteofdeficientfoundations—byrelyingheavilyonevermoresophisticatedexperiments.Ananecdotewillbeilluminating.Whenthisauthorwasanundergraduatestudentinthemidsixties,amoreseniorstudent(oritmighthavebeenaninstructor)toldhimthefollowing:“InEurope,westudytheoreticallyhowastabbreaks.Onethenbreaksthreestabstochecktheresultofthestudy.IntheUnitedStates,onebreaksthreethousandstabsandthenwritesdowntheruleaccordingtowhichtheybreak.”Eventually,EuropebecameliketheUSAinmattersofphysics,veryunlikewhatphysicswasthereinthefirstthreedecadesofthetwentiethcentury.Asformathematics,moneyhasalsodistorteditsdevelopment.AdifferentialgeometerbasedintheUnitedStatesofAmericaoncetoldme:“IdoRiemanniangeometrytoobtainfundingbut,inmysparetime,IdoFinslergeometry,whichiswhatIlike.”Infoundationsofphysicsonequestionssomeofitsmostbasicassumptions.Virtuallyallbutnotallpresentassumptionswillsurvive,butitisalmostanath-ematoquestionanyoneofthem,includingcertainlytheone(s)whichonedaywillberecognizedashavingbeingwrong.Whatdoesithavetodowiththisbook?Ifoneknewbettergeometryandcalculus,manybasicphysicsresultscouldbegivenbetterjustificationthanatpresent.Sincethisisabookondifferentialgeometry,letmestartwithsomethingthatpertainstogeneralrelativity(GR).

268CHAPTER1.ORIENTATIONSNowadays,GRincorporatestheLevi-Civitaconnection(LCC)[52],whichistheconnectioncanonicallydeterminedbythemetric.Itdidnotatonetime,sinceGRandtheLCCwerebornin1915and1917respectively.UndertheLCC,thereisnoequalityofvectorsatadistanceinGR.Itisthennotsurprisingthatwedonotyetknowwhatistheexpressionforgravitationalenergyand,relatedtoit,whatformwouldthecorrespondingconservationlawtake.Sufficetosaythatitismeaninglesstoaddthetinyenergy-momentumvectorsatsmallspacetimeregionsinordertogettheenergy-momentumforawholeregion.Allsortofnonsensehasbeenspokeninordertosidestepthisbasicproblem,asifitwereaninescapablefeatureofanygeometrictheoryofgravity.TheproblemlieswiththeadoptionoftheLCCbyGR.Einsteinunderstooditwhenhepostulatedthesocalledteleparallelisminhisattemptataunifiedtheoryinthelatenineteentwenties[39],buthenewverylittlegeometryandcalculusforwhatwouldberequiredtodevelophispostulate.Asseriousistheproblemwithelectromagnetic(EM)energy-momentum.Thetitleofsection4ofchapter27ofthesecondbookintheFeynmanLecturesonPhysics[40]reads“Theambiguityofthefieldenergy”.Inthatsection,speakingoftheenergydensity,u,andthePoyntingvector,S,RichardFeynmanstated:“Thereare,infact,aninfinitenumberofdifferentpossibilitiesforuandS,andsofarnoonehasthoughtofanexperimentalwaytotellwhichoneisright.”Furtherdownthetextheinsistsonthesubjectasfollows:“Itisinterestingthatthereseemstobenouniquewaytoresolvetheindefinitenessinthelocationofthefieldenergy”.Continuingwiththeargument,herepeatsasimilarstatementwithregardstoexperimentalverification:“Asyet,however,noonehasdonesuchadelicateexperimentthatthepreciselocationofthegravitationalinfluenceofelectromagneticfieldscouldbedetermined.”ConsiderontheotherhandthefieldtheoreticderivationoftheEMenergy-momentumtensorthroughvariationsoftheLagrangianpertainingtospacetimehomogeneity.Oneobtainswhatisconsideredtobethe“wrong”EMenergy-momentumtensor.The“right”oneisobtainedinLandauandLifshitzClassicalFieldTheory[51](andineverybody’stextwhousesthesameapproachtoEMenergy-momentum)byaddingtothewrongoneaspecifictermwhoseintegraloverallofspacetimeiszero.But,indoingso,oneischangingthegravitationalfieldandthus,inprinciple,thedistributionofgravitationalenergy.ViewedinthecontextoftheargumentbyFeynman,havingto“cheat”tocorrecttheresultofacanonicalargumentimpliesthattheLagrangianapproachcannotbetotallytrusted,orthattheelectromagneticLagrangianiswrong,orthatthevariationalmethodcannotbereliedupon,orsomeotherequallytroublesomeconsequence.Wewillnotwasteourtimewithrefutationsofhypotheticalalternativederivationsofwhatisconsideredtobetherightenergy-momentumtensor,sincewebelievethatonecandomuchbetter.Wethinkthattheproblemofgravitationalandelectromagneticenergiesareinextricablytied,andsoaretheproblemsrelatedtogravitationalphysicsmentionedearlierinthissection.Thesolutionthathasnotbeentriedisteleparallelism,whichentailsconnectionwithzeroaffinecurvature(bettercalledEuclideanorLorentziancurvature)on

271.5.MATHEMATICALVIRUSES9thestandard“metric”.Wenowenterthemostegregiousexampleofinsistingonthewrongmathe-maticsforphysics,thoughignorancemaylargelyexcuseit.YoungmindswhoaspiretoonedaychangetheparadigmputtheirsightsinYang-Millstheory,orstrings,orsomethingwith“super”atthefront,butnotintangentbundlerelatedgeometry.Howwouldyouachieveunificationwiththelatter?Assumethatacommonlanguageforboth,quantummechanicsandgravita-tion,didexist.Oneshouldstartbyusingthesamelanguageinbothcases.Hereisthegreattragedy:thatlanguageexists,namelyK¨ahlercalculusofdifferentialformsandconcomitantquantumtheory.InspiteofthemanyinterpretationalproblemswiththeDiractheory,itkeepsbeingusedeventhoughK¨ahler’stheoryiseasiertointerpretanddoesnothaveproblemslikenegativeenergysolutions,hasspinatparwithorbitalangularmomentum,etc.[46],[47],[48].Seealso[83].AgoodunderstandingofK¨ahlertheorytogetherwithanapproachtoGRwhereoneadoptsRiemanniangeometryformetricbutnotforaffinerelationsmaygoalongwaytowardsmakingGRandquantumphysicscometogether.Seechapter13.Letusmentionanavenuethatwehavenotexploredandthat,therefore,willnotyetbeconsideredinthisbook.TheHiggsfieldisthedifferencebetweenacovariantderivativeandaLiederivative[54].Inspiteofitsbrevity,ourbrieftreatmentofLiedifferentiationinthisbookwillallowustounderstandthatLiederivativesaredisguisedformsofpartialderivatives.These,aswellasexteriorderivatives,differfromcovariantderivativesinconnectionterms,whichsug-geststhepossibilitythatonemaybeundulyreplacing,intheargumentswheretheHigssfieldisconcerned,onetypeofquantityforanother(Differentderiva-tivesresultfromapplicationofacommonoperatortoquantitiesofdifferentvaluedness).Ifwemisrepresentonevaluednesswithanotherindescribingsomephysicalmagnitude,weshallbegettingthewrongderivatives.ThedetectionofaHiggsparticlemightbeinprincipletheexperimentalisolationofatermintroducedtocorrectanotherwisedefficientdescription.Inotherwords,anexperimentmightperhapsbeisolatingadominantenergytermthatrepresentsthatcorrection.1.5MathematicalVIRUSESThephysicistDavidHesteneshasdevotedhislifetothecommendabletasksofpreachingthevirtuesandadvantagesofCliffordalgebraovervectorandtensoralgebraandofapplyingittophysics.Hehashadthecourageof(anddeservescreditfor)advancingtheconceptofvirusesinmathematics,andforsuggestingwhat,inhisopinion,aresomeofthem[44].Hisdefinitionofmathematicalvirusis:“apreconceptionaboutthestructure,functionormethodofmathematicswhichimpairsone’sabilitytodomathematics”.Needlesstosaythatifpracti-tionerAseesavirusinpractitionerB,thelattermayretortthatitisAwhohasavirus.Weenumerateinstancesofwhatthisauthorconsiderstobeviruses.Wegivethemnames.Thebestantidoteagainstvirusesistofollowthegreatest

2810CHAPTER1.ORIENTATIONSmasters,whoclimbedthegreatestpeaksbecausetheypresumablylivedinthehealthiestmathematicalenvironments.BachelorAlgebraVirus.Itconsistsintryingtodoeverythinginjustonealgebrawhentwoormoreofthemarerequired.Forinstance,f(x,y,z)dx+g(x,y,z)dybelongstoonealgebra,andf(x,y,z)i+g(x,y,z)jbelongstoanotheralgebra.Thetensorcalculusisonewherethefundamentallydifferentnatureofthosetwoalgebrasisnottakenintoaccount.Itisabacheloralgebra.Therearetwoalgebrasinvolvedindxi+dyj+dzk.Theycomeintertwined,buttheresultisnotequivalenttoasingleoneofthem.ThisvirusistotallyabsentintheworkifE.K¨ahlerandE.Cartan.´UnisexVirus.Thisisamildvirusconsistinginthatasymbolisusedfortwodifferentconcepts.Forinstance,itiscommonintheliteraturetodefinedifferentialformsasantisymmetricmultilinearfunctionsofvectorfields.Fine.Butonecannotthensayatthesametime,asoftenhappens,thattheyareintegrands,sincer-integrandsarefunctionsofr-surfaces.Eitheroneortheother,butnotbothatthesametime.Itiscorrect(actuallyitiscustomary)tousethetermdifferentialformsforthefirstconcept,andthenusethetermcochainforthesecondone.Thereisnotaunisexvirusthere.Wedoprefer,however,tofollowW.Rudin,E.K¨ahlerandE.Cartaninreferringtocochains´asdifferentialforms.TransmutationViruses(mutationsoftheunisexvirus).RecallwhatwesaidintheprefaceabouttwotypesofsubscriptsintheK¨ahlercalculus.Oneofthemisforcovarianttensorvaluedness,theotheronebeingfordifferentialforms,abstractionmadeoftheirvaluedness.Applyingtoquantitieswithonetypeofsubscriptthedifferentiationthatpertainstotheothertypeisanexampleoftransmutationvirus:aquantityismadetoplayarolethatisnotinitstruenature.Letusbespecific.Wecanevaluate(meaningintegrate)differential1-formsoncurvesofamanifoldindependentlyofwhattheruletocomparevectorsonthemanifoldis.Butwecannotcomparevectors(whethercontravariantorcovariant)atdifferentpointsofthecurvewithoutaconnection.Anotherexampleofthetransmutationvirusconsistsinthinkingoftheelectromagneticfield—whichisafunctionof2-dimensionalsubmanifoldsofspacetime—asa(tangent)tensor.AsCartanpointedout,Maxwell’sequa-tionsdonotdependontheconnectionofspacetime[12]IfMaxwell’sequationswereviewedaspertainingtoanantisymmetrictensorfield,Maxwell’sequationswoulddependontheconnectionofspacetime.MonocurvatureVirus.Itconsistsinignoring(orwritingorspeaking)asifmanifoldsendowedwithaEuclideanconnection(i.e.metric-compatibleaffineconnection)hadjustonecurvature.BooksdealingwithEuclideanconnectionsusuallyfailtomakethepointthatthemetricdefinesametriccurvature,whichplaysmetricrolesindependentlyofwhattheaffineconnectionofthemanifoldis.Ifthelatter’saffineconnectionistheLCC,thesamecurvaturesymbolsthenplaytworoles,metricandaffine.This,however,hastheeffectofleavingreadersunawareofthisdoublerepresentation.Thatisaperniciousvirus.ItaffectedEinsteinwhenheattemptedunificationthroughteleparallelism[39].RiemannitisVirus.ThisvirusconsistsinviewingRiemanniangeometry

291.6.FREQUENTMISCONCEPTIONS11asthetheoryoftheinvariantsofaquadraticdifferentialforminnvariableswithrespecttotheinfinitegroupofanalytictransformationsofthosevariables.ElieCartanreferredtotheRiemanniansp´acesviewedfromthatperspectiveasthe“falsespacesofRiemann”(seepage4of[13]andmisconception(b)innextsection).Thisvirushasitsrootinthefailuretorealizethat,ashepointedoutinhisnoteof1922OntheequationsofstructureofgeneralizedspacesandtheanalyticexpressionofEinstein’stensor[10],“thedsdoesnotcontainallthegeometricrealityofthespace...”TheRiemannitisvirushasgivenrisetoaplaguethatinfectsvirtuallyallphysicistsandmathematicianswhodealwiththisgeometry.ItcausesamentalblockimpedingaframebundleviewofFinslergeometryandpotentiallysuccessfuleffortstounifythegravitationalinteractionwiththeotherones.CurveismVirus.ThisvirusisamutantoftheRiemannitisvirus.ItconsistsinviewingFinslergeometryasthegeometrybasedonanelementofarc,i.e.oncurves.TheFinslerspacessodefinedarefalsespacesinthesamesenseaswhenCartanreferredtotheoriginalRiemannianspacesasfalsespaces.Thisvirusdoesnotmakesickthosewhoworkonglobalgeometryorvariationalproblemssincetheyareinterestedinconnectionindependentresults.Butitcausesblindnessinthosewhoneedto(butfailto)seeotherconnectionswhichalsoarecompatiblewiththesameexpressionfortheelementofarc.InthespiritofCartan,Finslergeometryshouldbedefinedintermsofabundleofframes,orsomethingequivalenttoit,independentlyofdistances.Themetric,andthusthedistance,shouldthenbeviewedasderivedinvariantscausedbytherestrictionofthebundle.Thekeydifferencewithpre-Finslergeometryliesnotintheformofthemetric,butinthetypeoffibration.Ofcourse,geometryiswhatgeometersdoand,iftheywanttogothatpath,itistheirprivilege.ButtheirframesdonotthenfitthenatureofEinsteinelevators,ofgreatimportanceforphysicists.1.6FREQUENTMISCONCEPTIONSReadersshouldignorethecommentsinthissectionthattheydonotunderstand,whichmaybeallofthemfornewcomerstodifferentialgeometry.Iregretifsomeexpertsmayfindsomeofthesecommentstrivial.Judgingbywhatgoesintheliterature,oneneverknows.Thisbookisnotwhatreaderslikelyexpectjustbecauseofouruseofdiffer-entialformsandthemovingframemethod.Norisitbecause,inaddition,affinespaceistreatedseparately.Itisfirstandforemostanattempttomitigatetheeffectsofthehaphazardwayinwhichthemathematicaleducationofatheoreti-calphysicisttakesplace,whichisattherootofmisunderstandingspropagatinginliteratureondifferentialgeometryforphysicists.Examplesabound:(a)Doesagroupdetermineadifferentialgeometryintheclassical,i.e.(non-Yang-Mills)senseoftheword?(no,itdoesnot;apairofgroupandsubgroupsatisfyingacertainpropertyisneeded[27],[67]).Seealsodiscussioninsection4ofchapter12inconnectionwithKlein.

3012CHAPTER1.ORIENTATIONS(b)Istheinfinitegroupofcoordinatetransformationsoftheessence—i.e.adefiningproperty—ofclassicaldifferentialgeometry?(no,itisnot;asCartanputit,suchawayofthinkinginRiemanniangeometrydoesnotmakeevidentandactuallymasksitsgeometriccontents,intheintuitivesenseoftheterm[21]).Seealsosection10.1.(c)“Coordinatesarenumbers”(equivocalstatementthoughnottoodan-gerousinactualpracticeifoneusesthesametermandsymbolsforcoordinatefunctionsasfortheirvalues;ifthetermcoordinatesdenotedonlynumbers,theirdifferentialswouldvanish).Seesection2.1.(d)“Whenusingthemethodofthemovingframeusingdifferentialforms,therearetwotypesofindicesintheaffine/Euclideancurvature”(wrong,becauseinadditiontothesuperscriptstherearetwodifferenttypesofsubscripts).Seeprefaceandsection2.1.(e)“dr(=dxi+dyj+dzk)isa(1,1)−tensor”(thisisanincorrectstatementfromtheperspectiveoftheCartancalculusanditsgeneralizationknownastheK¨ahlercalculus;itisrathera(1,0)−tensor-valueddifferential1−form).Seesections2.1,5.2,5.7and8.2.(f)“Theωi’s,whicharecomponentsofdr(=ω1e+ω2e+ω3e)more123generalthanthedxi’sarenotdifferential1−formsbecausetheyarenotinvariantundercoordinatetransformationsbuttransformlikethecomponentsofavector”(theyare1−formsinthebundleofframesofEuclidean3-space,buttheirpull-backstosectionsofthatbundlearenot,sincetheythentransformlikethecomponentsofatangentvectorfield).Seesections5.8and6.1.(g)“Anaffinespaceisavectorspacewherewereplacepointswithvectors”(notuntilwechooseapointintheformertoplaytheroleofthezerovector,sinceaffinespacedoesnothaveaspecialpointthatwecouldcallthezero).Seesection5.1.(h)”Skew-symmetriccovarianttensorsconstituteasubalgebraofthetensoralgebra”(wrongsincetheydonotconstituteaclosedsetunderthetensorproduct;theyratherconstituteaquotientalgebra).Seesections1.3and3.6.(i)“3-dimensionalEuclideanspacedoesnothaveaconnectiondefinedonit”(wrong,ithasthetrivialonewhereasectionofconstantbases(i,j,k)exists,i.e.equaltothemselveseverywhere;ifonehadmadeadifferentchoiceforthecomparisonofvectors,thespacewouldnolongerbeEuclidean).Seesections6.1and7.2.1.(j)“ThecanonicalconnectionofaspaceendowedwithametricisitsLevi-Civitaconnection”(notnecessarilytrue,asthetorushasthenaturalconnectionwherethecirclesresultingfromintersectingitwithhorizontalandverticalplanesarelinesofconstantdirection;thisisnotitsLevi-Civitaconnection,whichiscanonicalofthemetric,butnotofthemanifolditself).Seesection7.4.(k)Isametriccompatibleaffineconnectionactuallyanaffineconnection?(no,itisratheraEuclideanconnection,whichlivesinthesmallerbundledefinedbytheEuclideangroup).Seesections7.2,9.1and9.2.(l)IsthesocalledaffineextensionofanEuclideanconnectionbythelineargroupanactualaffineconnection?(no,itonlylooksthatwaysincetheframebundlehasbeenextended;butthenumberofindependentcomponentsofthe

311.6.FREQUENTMISCONCEPTIONS13connectionremainswhatitwas,notthesameasforaffineconnections).Seesections6.2and6.6.(m)“Thereissuchathingastheteleparallelequivalentofgeneralrelativity”(wrong,thegeneraltheoryofrelativitythathadnotyetacquiredtheLevi-Civitaconnectiondidnothaveanyconnectionand,forlackofgreatergeometriccontents,wasnotequivalenttoatheorycontaininganaffineconnection;andthegeneraltheoryofrelativitywithLevi-Civitaconnectioncannotbeequivalenttoonewithteleparallelism,sincethefirstonedoesnotallowforequalityofvectorsatadistanceandthesecondonedoes;asanexampleofthedifference,sufficetosaythatonecannotsayatthesametimethatsomespaceisandisnotflat).Seeremark(v)andsections7.2to7.4.Seealsosection8.10fortheimpactonthetypeofconservationlawthatwemay(ormaynot!)havedependingonthetypeofconnection.(n)Istheexteriorderivativeofascalar-valueddifferentialformacovari-antderivative?(itcertainlyhastherightcovarianttransformationproperties,thoughtheconceptdoesnotrespondtowhatgoesbycovariantderivativeinthetensorcalculus).Seesection6.3.(o)Istheconservationlawofvector-valuedquantities,likeenergy-momentum,dependentonthebehaviorofpartialderivatives,orofcovariantderivativesinthesenseofthetensorcalculus?(noneofthetwo,asitdependsonthebehav-iorofwhatCartanandK¨ahlerrefertoasexteriorderivativesofvector-valueddifferentialforms,moreoftencalledexteriorcovariantderivatives;and,yet,noteventhisrefinementisquitecorrectasweshallshow).Seesection8.10.(p)Doestheannulmentoftheexteriorcovariantderivativeimplyaconser-vationlaw?(onlyiftheaffine/Euclideancurvatureiszero,foronlythencanoneadd–readintegrate–thelittlepiecesof“conserved”vectorortensoratdifferentpoints;noticethattheconservationlawofthetorsionisgivenbythefirstBianchiidentity,which,inadditiontotheexteriorcovariantderivativeofthetorsion,containstheaffine/Euclideancurvature).Seesection8.10.(q)Isitpossibletospeakofa(1,3)−tensorcurvature?(yes,a(1,3)−tensorwithgeometricmeaningthathasthesamecomponentsastheaffinecurvatureexists,butasaconceptdifferentfromthestructuralconceptofcurvature,whichisa(1,1)−valueddifferential2−form).Seesection8.6.(r)Doesthe(1,3)natureofthetensorin(q)makeitavector-valueddiffer-ential3−form?(no;inadditiontothereasonalreadymentionedin(h),thereisnottotalskew-symmetrywithrespecttothethreesubscripts).Combinecontentsofsections5.7and8.6.(s)IstheRiemanniancurvatureanaffine(or,saidbetter,Euclidean)curva-ture?(notnecessarily,asitneednotplay—anddidnotplayforhalfacentury—anyaffineorEuclideanrole;itwasconceivedbyRiemannforanotherrole).Seesections7.5,9.8and10.1.(t)ThequantitiesthatappearintheequationsofstructureunderthenamesofaffineandEuclideancurvatures,aretheytensorsofrankfourorvector-valued3−forms?(noneofthetwo;fromasection’sperspective,theyare(1,1)−tensor-valueddifferential2−forms,andtheyareLiealgebravaluedfromtheperspectiveofbundles).Seesections5.7,8.13and9.2.

3214CHAPTER1.ORIENTATIONS(u)InRiemanniangeometry,canweintegratetheenergy-momentum“ten-sor”?(no,asimpliedin(p);however,atatimewhenRiemanniangeometrydidnothaveaconceptofaffinecurvaturebutonlyofmetriccurvature,onewouldbeentitledtodomanyoftheintegrationsthatgeneralrelativistspresentlydosince,withoutknowingit,theunstatedbutimplicitassumptionthattheaffinecurvatureofspacetimeiszerowouldjustifythosecomputations).Seesections8.10and9.8.(v)InEinstein’sgeneraltheoryofrelativityasof1916,wastheRiemanniancurvaturetheonedeterminingthechangeofavectorwhentransportedalongaclosedcurve?(wrongstatement,astheconceptofvectortransport,whichisanaffineconcept,didnotexistinthemathematicalmarketuntiloneyearlater;theRiemannian“symbols”werethenseenasalsoplayinganaffinerole,butthiswasnotanecessarycourseofactionforgeneralrelativity,sinceonecouldhaveextendedtheoldRiemanniantheorywithanaffineconnectionotherthanLevi-Civita’s).Gotoremark(m)andseealsosections10.1to10.3.(w)Isthetorsiontheantisymmetricpartoftheconnection?(no,notingeneral;thestatement,whichactuallyreferstothecomponentsofthetorsion,failstobecorrectinnon-coordinatebasisfields).Seesection8.2.(x)“Torsionisrelatedtospin”(incorrectstatement,asthetorsionhastodowithtranslationsandspinhastodowithrotations[59]).Seesection8.1and,inmoreelaborateform,see[88].(y)“ThecyclicpropertyofRiemann’scurvature,whenconsideredasthecurvatureofanEuclideanconnection,issimplyonemoreofitsproperties”:(wrong,itisthedisguisedformthatthefirstBianchiidentitytakeswhentheexteriorcovariantderivativeofthetorsioniszero,and,inparticular,whenthetorsionitselfiszero).Seesections8.11and10.3.(z)“Usingconnections,onecandeveloponflatspacescurvesofnon-flatspaces.Doclosedcurvesonnon-flatspacesdevelopontoclosedcurvesonflatspacesifthetorsioniszero?”(no!,thosedevelopmentsfailtocloseingeneral,regardlessofwhetherthetorsioniszeroornot;thedifferencethatthetorsionmakesisthat,asclosedcurvesaremadesmallerandsmaller,theirdevelopmentstendtowardsclosedcurvesmuchfasterwhenthetorsionvanishesthanwhenitdoesnot).Seesection7.3.3and7.4.1;onthesmallnessofcontours,see8.9.1.7PrerequisiteandanticipatedmathematicalCONCEPTSFornon-expertreaders,weproceedtobrieflymentionorexplainwhatstructuresareinvolvedinthisbook.Ifthebriefremarksgivenherearenotenough,nothingislost;readerswillatleasthaveastartingpointfortheirconsultationsinbooksonalgebra,analysisorgeometry.Looselyspeaking,aringisasetofobjectsendowedwithtwooperations,calledadditionandproduct.Theringisagroupunderadditionandhas,there-fore,an“inverseunderaddition”,calledopposite.Amongtheringsthereare

331.7.PREREQUISITE,ANTICIPATEDMATHEMATICALCONCEPTS15thefields,likethefieldofrealsandthefieldofcomplexnumbers.Afieldhasmorestructurethanageneralring,namelythateachelementexceptthezero(i.e.theneutralelementunderaddition)hasaninverse.Functionsconstituterings,butnotfields,sinceafunctiondoesnotneedtobezeroeverywhere(thezerofunction)inordernottohaveamultiplicativeinverse.Andtherearestructureswhich,likeavectorspace,involvetwosetofobjects,saythevectorsontheonehand,andthescalarsbywhichthosevectorsaremultiplied,ontheotherhand.Theothermostcommonstructuresinvolvingtwosetsofobjectsaremodulesandalgebras.Althoughonefirstdefinesmodulesandthenvectorspacesasmodulesofaspecialkind,generalfamiliaritywithvectorspacesrecommendsadifferentperspectivefornon-expertreaders.Amoduleislikeavectorspace,exceptforthefollowing.Inavectorspacewehavethevectorsandthescalars,thelatterconstitutingafield.Inamoduleweagainhavethe“vectors”,butthescalarsconstituteonlyaring,likearingoffunctions.Wemay,inadvertentlyornot,abusethelanguageandspeakofvectorspaceswhenweshouldspeakofmodules.Finally,analgebraisamodulethatisalsoaring.Inanticipationofmaterialtobeconsideredinchaptertwo,letusadvancethatthedifferentialformsofagivengrade(saydifferential2−forms,i.e.inte-grandsofsurfaceintgrals,whichareofgrade2)constituteamodule.Andthesetofalldifferentialformsofallgrades(togetherwiththoseofmixedgrade)builtuponagivenspaceoruponamoduleofdifferential1−formsconstituteanalgebra,exteriorandCliffordalgebrasinparticular.Anotherimportantalgebraicconceptinthisbook,thoughitwillmakeonlybriefappearance,istheconceptofideal.Normallyonespeaksofanidealwithinanalgebra,buttheconceptisextendabletoanysetwherethereisamultipli-cationproduct.AleftidealisasubsetIofasetSsuchthatmultiplicationsofelementsofIbyelementsofSontheleftalwaysreturnselementsinI.Itmustbeobviousnowwhatarightidealis.Assimpleexamples,letusmentionthatintegersdonotmakeanidealwithintherealnumbers,neitherontherightnorontheleft,sincetheproductofanintegerandarealnumberisnotanintegernumber.Theevennumbersconstituterightandleftidealswithintheintegerssincetheproductofevenandintegeriseven(countingzero).Idealsareimportantinquantummechanics,notsomuch,ifatall,instan-darddifferentialgeometry.But,sincethisauthoraspirestobringdifferentialgeometryandquantummechanicsclosertoeachother,readerswillfindaninklingabouttheproximitytogeometryofphysicallyrelevantidealsinthelastsectionofchapter13.Letusnowdealwithanissueclosertoanalysis.Wearegoingtoberathercasualwithouruseofthesimilarconceptsofmap,functionandfunctional.Fortherecordandsothatanybodywhowantsmaymakethetextmorepreciseinthisregard,wedefinethosethreeconcepts.Amapisanapplicationfromanarbitrarysettoanotherarbitraryset.Itcanthusbeusedinallcaseswherethemorespecifictermsoffunctionandfunctionalalsoapply.Thedomainofmapscalledfunctionsalsoisarbitrary,buttherangeorcodomainmustbeatleastaring.Finally,afunctionalmapsanalgebraoffunctionstoanotheralgebra,or

3416CHAPTER1.ORIENTATIONStoitselfinparticular.Whatfollowswillbepresentedformallyinlaterchapters.Alittleincursionatthispointmaybehelpful.WherethetermsaffineandEuclideanarecon-cerned,wemaybedealingwithalgebraorwithgeometry.AEuclideanvectorspaceisavectorspaceendowedwithadotproduct.Thatisalgebra.Anaffinespaceisaspaceofobjectscalledpointssuchthat,afterarbitrarilytakingapointtoplaytheroleofzero,wecanestablishaone-to-onecorrespon-dencebetweenitspointsandthevectorsofavectorspace,calleditsassociatedvectorspace.IfthevectorspaceisEuclidean,theaffinespaceiscalledEuclideanspace(i.e.without“vector”betweenthewords“Euclidean”and“space”).ThestudyofaffineandEuclideanspaceisgeometry.Accordingtotheabove,apointseparatesalgebrafromgeometry.Inalgebrathereisazero;ingeometry,thereisnotanybutweintroduceitarbitrarilyinordertobringalgebraintoit.Thereareadditionalsubtledifferences.Theyarerelatedtowhatwehavejustsaid.Theyhavetodowithwhatisitinalgebrathatalgebraistsandgeometersconsidermostrelevant.Grosslyspeaking,itismatricesandlinearequationsforalgebraists.Anditisgroupsforgeometers.Moreonthisinsection9ofchapter6.RatherthanviewingaffinegeometryasageneralizationofEuclideange-ometry,whichitisinsomesense,itisbettertothinkintermsofthelatterbeingarestrictionoftheformer.Hereiswhy.Anythingaffineisrelatedtothecomparisonofvectorsandtheirframes.Therestrictionenrichesthecontentswithaninvariant,thedistance.ItisinvariantundertheEuclideangroup,notundertheaffinegroup.Themaintopicofthisbookisadifferenttypeofgeneralization,attributabletoCartan.AffineandEuclideangeometriesareKleingeometries(thinkflatgeometriesforthetimebeing).The“generalizedspaces”ofthetypetowhichRiemanniangeometrybelongsarecontinuousspacesendowedwithEuclideanconnections,improperlyknownasmetric-compatibleaffineconnections.Forthefirstsixdecadesofitslife(sincemidnineteenthcentury),aRiemannianspacewasviewedasageneralizationofEuclideanspaceinthesenseofhavingamoregeneralexpressionfordistance.Buttherearegeneralizationswhich–ratherthanthedistanceorinadditiontoit—generalizetheaffinecontents.TheaffinecontentsoftheoriginalRiemanniangeometrystartedtobeconsideredonlysixdecadesafteritsbirth.Cliffordalgebraisanalgebraicconceptlargelyoverlookedbygeometers,thoughnotbycertainalgebraistsandmathematicalphysicists.ThealgebraofintroductorycollegecoursescannotbeextendedtoEuclideanspacesofhigherdimensionsandcannot,therefore,beproperlycalledEuclideanalgebra.SocalledCliffordalgebraisEuclideanalgebraproper.Abriefincursionintoitisgivenattheendofchapterthree.Thereareseveralbooksthatdealwiththatalgebra.Weshallnottakesidesandmentionanysincewewouldalsospeakofitslimitations.Becauseoftradition,onestudiesEuclideangeometriesandtheirCartangeneralizationswithtoolsthatpertaintoaffinegeometry,i.e.linearalgebraandmatrices.ButonecouldstudyEuclideangeometrywithtechniquespecific

351.7.PREREQUISITE,ANTICIPATEDMATHEMATICALCONCEPTS17toit,i.e.Cliffordalgebra.ItcouldrightlybecalledEuclideanalgebra,asitpertainstoanythingEuclidean(Forexperts,thefactthatwecanputasidepartofitsstructuralcontentsanddowithitprojectivegeometrydoesnotnegatethisstatementpreciselybecause“weareputtingasidestructuralcontents”).ThetermCartangeneralizationevokes(orcertainlyshouldevoke)thestudyofstructureofmanifoldsfromtheperspectiveoftheirframebundles,notofsectionsthereof(alsoknownasframefields).Whenyoudothat,LiealgebrasjumptotheforethroughtheLiegroupsinvolvedinthedefinitionofthosebundlessinceaconceptofdifferentiationofaLiegroupisinvolvedindefiningaLiealgebra.Weshallthusmakethefirstacquaintancewiththisconceptattheendofchapter4.Keyexamplesofthemwillemergenaturallyinlaterchapters.Thefollowingremarkonstructureisrequiredbecauseofhowclassicaldiffer-entialgeometryisdealtwithinbooksonthegeometryofphysics.Aspointedoutinthepreface,bundlesandLiealgebrasarerarelyusedforintroducingtheconceptofconnectioninclassicaldifferentialgeometry,unlikeinYang-Millsthe-ory.This“lackofusage”virtuallyimpedestherealizationofthefollowingthat,aspointedoutby(FieldsMedalist)DonaldsonandKronheimerintheopeningparagraphofchapter2oftheirbook[35]“ThekeyfeatureofYang-Millstheoryisthatitdealswithconnectionsonauxiliarybundles,notdirectlytiedtothegeometryofthebasemanifold.’Thusitisnotamatterofwhethertherearegaugetransformationsornotinclassicaldifferentialgeometry.Thereare,sincethosetransformationsaresimplychangesofframefield.Hereisaquestionthatalmostnobodyraisesnowadays:coulditbethatthoseauxiliarybundlesthatarepresentlyconsideredtobeoftheessenceofYang-Millstheoryaredisguisedformsofsomesophisticatedbundlesdirectlyrelatedtothetangentbundle?PhysicistshavenotseenaproblemwithviewingSU(2)inthatway.ButtheycannotseeSU(3)inthesameway.Thatistheproblem.ApparentlyyoucannotmixanSU(2)fromthetangentbundlewithSU(3)fromanauxiliarybundle.Well,theanswerissimple.OnemayviewSU(3)alsofromatangentbundleperspective.Oneneedsforthatastillbetterknowledgeofdifferentialformsthanwhatyouaregoingtogetinthisbook.Sostartwiththisbook,sinceyouwillfirstneedtoknowitscontents.Laterpublicationsbythisauthorwilllargelyanswertheopeningquestionofthisparagraph.

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39Chapter2DIFFERENTIALFORMS2.1AcquaintancewithdifferentialformsThemostgeneralobjectsthatwehavetodealwithinthisbookaretensor-valueddifferentialforms.Thesimplestonesarethescalar-valuedones,orsimplydifferentialforms.Theliteratureonthissubjectisfraughtwithdangersinthedefinitionsandtheirimplications.Forinstance,onemayencounterthestatementthatadifferentialformisacovariantskew-symmetrictensor.Ifthisisthecase,whatisatensor-valueddifferentialform?Isitperhapsacontravariant-tensor-valuedcovariantskew-symmetrictensor?Ishallnotpursuethislineofargumentfurthertoavoidconfusingreaders.Weheremakesomeacquaintances.A(scalar-valued)differential1−formisthedifferentialofafunction,whichcantakeformsassimpleasdx,oras2dx+3dy−7dz,(1.1)and,moregenerally,asfdx1+gdx2+hdx3+...,(1.2)wherethexiarearbitrarycurvilinearcoordinatefunctionsinsomegeneral-izedspace(indifferentialgeometry,thegeneralizedspacesofinterestreceivethenameofdifferentiablemanifolds).Thecoordinatesshouldnotbeneces-sarilyCartesian,andthespacenotnecessarily3-dimensional.Thef,g,...arefunctionsofthosecoordinates,sotheyarefunctionsoffunctions,i.e.compositefunctions.Thecomponentsofthesedifferential1−formsaresaidtobe(2,3,−7)and(f,g,h,...)relativetobasesofdifferential1−formsinrespectivespaces.Wehavespokenofcoordinatesasfunctions.Tobespecific,thexofthesetof(x,y,z)isafunctionwhichassignstoapointof(say)aEuclideanspaceanumber.Thesymbolxalsoisusedtodenotethevaluetakenbythatfunctionatthatpoint.Weshallusethetermcoordinatestorefertoboth,suchfunctionsandthevaluesthattheytake,theappropriatemeaningineachinstancebeingobviousfromthecontext.21

4022CHAPTER2.DIFFERENTIALFORMSThereasonforsometimesusingthequalificationscalar-valuedistomakeclearthat,whensayingit,weareexcludingfromourstatementsdifferentialformslikedxi+dyj+dzk.(1.3)Thisisavector-valueddifferential1−form.Ifweusethesymbolsxl(l=1,2,3)toreferto(x,y,z),andaltoreferto(i,j,k),wemayrewritethisvector-valueddifferential1−formasl=3dxla=dxla,(1.4)lll=1wherewehaveusedEinsteinsummationconvention(Thisconventionconsistsinimplyingasummationsigninfrontofexpressionswithrepeatedindices,oneoftheindicesbeingasuperscriptandtheotherbeingasubscript).If,inaddition,mmmδlisthesocalledKroneckerdelta(i.e.δl=1,ifl=m,andδl=0,ifl=m),wemayrewrite(1.3)asδmdxla,(1.5)lmwhichexhibitsδmascomponentsintermsofthebasis(dxla)ofvector-valuedlmdifferential1−forms.Wemustemphasizethat(1.3)isnota(1,1)tensor,sincedifferentialformsarenottensors.Afunctionwhich,likeF(x,y,...),(1.6)exhibitsnodifferentialsisascalar-valueddifferential0−form.Similarly,xi+yj+zk(1.7)isavector-valued0−form,equivalentlyavectorfield.Readerswhohaveprob-lemsseeingthisasavectorfieldbecauseallthevectors(1.7)“start”atthesamepointneedonlythinkofitasthenegativeofthevectorfield−(xi+yj+zk),withmaybeviewedasextendingfromalloverthespaceandendingattheorigin.Weshalllaterlearnwhythedxdyundertheintegralsignofsurfaceintegralsshouldbewrittenasdx∧dy,(1.8)wherethesymbol∧denotesexteriorproduct(soontobedefined),whichisthesamereasonwhythedxdydzthatwefindundertheintegralsignofvolumeintegralsshouldbewrittenasdx∧dy∧dz.(1.9)Thelasttwodisplayedexpressionsarereferredtoasscalar-valueddifferential2−formand3−formrespectively.Anotherexampleofdifferential2−formisady∧dz+bdz∧dx+cdx∧dy.(1.10)Thelikesofai∧j(1.11)

412.2.DIFFERENTIABLEMANIFOLDS,PEDESTRIANLY23arecalledbivectors.Theyaretovectorswhatdifferential2−formsaretodif-ferential1−forms.Somethinglikea(i∧j)dx(1.12)wouldbecalledabivector-valueddifferential1−form,which,byanabuseoflanguage,is(often)called2−tensor-valueddifferential1−form.Andonecouldalsohavevector-valueddifferential2−forms,etc.Exceptforindicationtothecontrary,thetermdifferentialformisreservedhereasinmostoftheliteraturefortheskew-symmetriconeswithrespecttoanypairofindiceswithinagivenmonomial(singleterms,asopposedtosumsthereof).Thetermsinthesumswillbeallofthesamegradeinthisbook.Weneedtodistinguishsuchdifferentialformsfromthosewhich,likethemetric,gdxidxj,arequadraticsymmetricdifferentials.ijOfgreatimportanceisthefactthatthetensorvaluednesscomesintwotypes,contravariantandcovariant.Theusualoneisthecontravariantone.Thecovariantoneisusuallyinducedbyandaccompaniestheformer.Sufficetoknowatthispointthatthosevectorsthateventhemostinexperiencedreadersarefamiliarwitharecalledcontravariant.Acovariantvectorisalinearfunctionofcontravariantvectors.Thenextparagraphcanbeskippedbyreadersbotheredbystatementscontainingtermswithwhichtheyarenotyetfamiliar.Tensor-valuedness(1,1)meansoncecontravariantandoncecovariant.Asalreadymentionedinthepreviouschapter,theaffinecurvatureisa(1,1)−valueddifferential2−form.Onecanassociatewiththeaffinecurvaturea(1,3)−valueddifferential0−form,orsimply(1,3)tensor.Itisnotadifferential3−form,amongotherreasonsbecauseitisnotskew-symmetricwithrespecttoitsthreesubscripts.Forlackofabettername,weshallcallitcurvaturetensor.Itplaysarelativelyminorrole,unlikeaffinecurvatureandtorsion,whichbelongtothecoreofgeometry.Warning:thenotation(p,q)hastobeviewedineachauthor’scontext,sinceitmayreferinsomeofthemtoadifferentialq−formthathasvaluednessp.Ofcourse,twoindicescannotplaytherolethatthethreeindicesin(s,t)−valuedr−formsplay.2.2Differentiablemanifolds,pedestrianlyThereisagreatvirtueincalculusofdifferentialforms.Itisappropriatefordifferentiationingeneralspacesknownasdifferentiablemanifolds,which,ingeneral,lackametricandaconnectionorruletocomparevectorsatdifferentpoints.Wemust,however,haveaconceptofcontinuityinthespaceinquestion,whicheliminatesautomaticallydiscretesetsofpoints.Thesetmustbesuchthatregionsofthesamecanberepresentedunequivocallybyopensetsofn-tuplesofrealnumbers(itcouldalsobecomplexnumbers,butnotinthisbook).Theopennessofthesethastodowiththebehavioratthebordersoftheregion.Wesaid“unequivocally”inordertodealwithsituationslikethefollowing.ConsidertheEuclideanplanepuncturedatonepoint,whichwetakeastheoriginofpolarcoordinates.Wecanputallitspointsinaone-to-onecorrespondence

4224CHAPTER2.DIFFERENTIALFORMSwithallthepairsofnumberssatisfyingthefollowingproperties.Oneelementofthepair,ρ,takesallpositiverealvalues.Theotherelementofthepair,φ,takesallnon-negativerealvalueslessthan2π.Butthefullplaneisnotcoveredbyasystemofpolarcoordinatessincetheoriginofthesystemiscoveredbyspecifyingjustρ=0.Wewouldneedanothersystemofcoordinatesthatwouldcovertheoriginofthefirstinordertoachievethateverypointoftheplanewillbecoveredbyatleastonecoordinatesystem.Itisforthisreasonthatonespeaksofsystemsoflocalcoordinates.Thetermregionwillbeusedinthisbooktodenoteopensubsetswiththesamedimensionasthemanifold.Thereasontooftenconsideraregionandnotthewholemanifoldisthatonemayneedtoavoidsomespecialpoint,liketheoriginofthepolarcoordinates.Acoordinatesystemthenisamapfromaregionofthedifferentiablemani-foldtothesetofn-tuples.Ifwechooseapoint,themapdeterminesann-tupleofcoordinatesofthepoint.Hence,wehavethecoordinatemapandthecoor-dinatesascomponents,orvaluestakenbythesetofcoordinatefunctionsthatconstitutethemap.Whentwocoordinatesystemsoverlap,itisrequiredthatthefunctionsexpressingthecoordinatetransformationarecontinuousandhavecontinuousderivativesuptosomeorder,appropriatetoachievesomespecificpurpose.Apairofregionandcoordinateassignmentiscalledachart.TheregioninthechartofCartesiancoordinatesintheplaneisthewholeplane.Andtheregioninthechartforthesystemofpolarcoordinatesistheplanepuncturedatthechosenoriginofthesystem.Theinverseofacoordinatefunctionwillcertainlytakeusfromthecoordi-natestothepointsonthemanifoldrepresentedbythosecoordinates.Letuscalltheminversecoordinatefunctions.LetmandnbetherespectivedimensionsofmanifoldsMandN,i.ethenumberofcomponentsneededtorepresenttheirpointsinthemannerprescribed.Intermsofcoordinates,afunctionfromamanifoldMtoanothermanifoldNisacompositeofthreefunctionsinsuc-cession.Thefirstonetakesonefromm-tuplestothemanifoldM.ThesecondoneisthegivenfunctionfromMtoNandthethirdonetakesusfromNton-tuples.WesaythatthefunctionfromMtoNisdifferentiableiftheforgoingcompositefunctionisdifferentiable.Achangeofcoordinatesona(regionof)amanifoldMistheparticularcasewhereM=N.Theidentityfunctiononthereallineassignsacoordinatetoeachofitspoints.Inotherwords,apointinthereallineisitsowncoordinate.ThustheexpressionintermsofcoordinatesofarealfunctiononM(i.e.arealvaluedfunctionorsimplyrealfunction)reducestothecompositionoftwofunctions,namelyfromthecoordinatestothepointonthemanifoldandfromthelattertoR1.Letfdenotethesetofthefithattakethevaluesxi.Theinversefunctionf−1takesusfromthecoordinatestothemanifold.Onedefinesthedifferen-tialdgofarealfunctiongonthemanifoldas(recallEinstein’ssummation

432.3.DIFFERENTIAL1−FORMS25convention)∂gidg=dx,(2.1)∂xiwhere∂g∂(g◦f−1)=.(2.2)∂xi∂xiItwillbeshowninthenextsectionthatdgisindependentofthecoordinatesystemchosenforthedefinition.Itisconvenienttointroducethenotationg(xi)forg◦f−1.TheexpressionfordggivenabovethenbecomesthedgofthecalculusinRn.Thenotationg(xi)reflectsthefactthatonenormallygivespointsPofamanifoldbygivingtheircoordinatesinsomeregion.Withthisnotation,equationslookasrelationsonRnratherthanasrelationsonthemanifold.2.3Differential1−formsTheintegrandsoftheintegralsonadifferentialmanifold(onitscurves,surfaces,etc.;allofthemoriented)constituteastructurecalledanalgebra.Tobemorespecific,itisagradedalgebra.Asafirstapproachtotheconceptofgradeinagradedalgebra,letussaythefollowing.Theintegrandsinintegralsoncurves,surfaces,volumes...aresaidtobeofgrade1,2,3,...Therealfunctionsaresaidtobeofgradezero(WeshalljustifythelaststatementwhendealingwithStokestheoreminchapter4).Thesubspacesconstitutedbyalltheelementsofagradedalgebrathathavethesamegradeconstitutemodules(generalizationoftheconceptofvectorspace).Forthepresentexpositorypurposes,letussaythatthealgebramaybeconsideredasbuiltfromthemoduleofelementsofgradeonebysomekindofproduct(exterior,tensor,Clifford).Inalgebrasofdifferentialforms,themoduleofgradeoneisthemoduleofdifferential1−forms.Inthisbook,theproductofscalar-valueddifferentialformsismainlytheexteriorproduct.InK¨ahler(barelyconsideredinthisbook),itwillbetheCliffordproduct.Themeaningoftheelementsinthealgebrawillbeinducedbythemeaningofthedifferential1−forms.Thesearelinearcombinationsofthedifferentialsofthecoordinates.So,whatisthemeaningofthedifferentialsofthecoordi-nates?Non-expertreadersareinvitedtocheckthisinthemathematicalbooksoftheirlibraries.Thecomparisonofdefinitionsisaveryinterestingexercise.Au-thorssometimesstateintheintroductionoftheirbooksthat,say,adifferential1−formisafunctionoforientedcurves(whoseevaluationistheirintegration),andinthemaintextitisalinearfunctionofvectorfields.Thisisunsatisfactorysincethesearedifferentconcepts.Onecanspeakofaexterior(i.e.Cartan’s)calculusforbothconceptsofdifferentialforms.Intheopinionofthisauthor,itisbettertothinkofthemasintegrandsifonewantstounderstandCartan’sapproachtodifferentialgeom-etrywell.ThisisnolongeramatterofopinionintheK¨ahlercalculus,wherecomponentscanhavetwoseriesofsubscripts(andoneofsuperscripts).Thetwotypesofsubscriptscorrespondtotwodifferenttypesofobjectsandassociated

4426CHAPTER2.DIFFERENTIALFORMSdifferentiations.Onetypefitsnaturally(inthecaseofjustonesubscriptofeachtype)thefunctionsofcurvesandtheotheronethelinearfunctionsofvectorfields.InRudin’sbook[65]inanalysis,(scalar-valued)differentialformsareintroducedasintegrandsinthemaintext,ratherthanjustpayinglipservicetothemintheintroductionofthebookandthenbeingignored.Regardlessofhowdifferentialformsaredefined,itisthecasethatx=bdx=b−a,(3.1)x=aordx=xB−xA,(3.2)cwherecisanorientedcurvewithAandBasoriginandendpoints.Thatisforusthedefinitionofdx.Itissimplyaconsequenceofthedefinitionofadifferential1−formasamap(orfunction)thatassignstoeachopencurveconamanifoldtheintegrala(x)dxi.(3.3)icThemapmaybeexpressedincoordinateindependentformasaformallin-earcombinationofdifferentialsofsmoothfunctionswithsmoothfunctionsascoefficients12mf1dg+f2dg+...+fmdg.(3.4)Heremisanypositiveinteger.Itcanbeequalto,greaterthan,orsmallerthann.Atthispoint,thepositionoftheindicesin(3.4)isnotofgreatimportance.Whenonedevelopsthedgiaslinearcombinationsofdifferentialsofcoordi-nates,oneobtainsamorepracticalexpression.Sincedgidgi=dxj,(3.5)dxj(i=1,...,m,j=1,...,n),(3.4)becomes∂giα=fdxj=a(x)dxj,(3.6)i∂xjjwhere∂giaj(x)=fi.(3.7)dxjTwoarbitrarydifferential1−formsαandβ,α≡a(x)dxi,β≡b(x)dxi,(3.8)iiaresaidtobeequalonsomesubsetUofthemanifold(curve,surface,...,region)ifai=bii=1,...,n(3.9)

452.3.DIFFERENTIAL1−FORMS27atallpointsofthesubset.Wedefineanassociativesumofdifferential1−formsbyiα+β+γ...=(ai+bi+ci...)dx,(3.10)andtheproductbyafunctionfasifα=faidx.(3.11)If(x1...xn)constitutesasetofnindependentvariables,theset(dx1...dxn)constitutesabasisfordifferential1−forms.AchangeofcoordinatesinducesachangeofsuchbasesattheintersectionofthecorrespondingsetsU1andU2ofM.Letusreferthesamedifferentialformαtotwodifferentcoordinatesystems.Wehavejiα≡ajdx=aidx.(3.12)Sincejjiidx=(∂x/∂x)xdx,(3.13)itfollowsbysubstitutionof(3.13)in(3.12)andcomparisonofcomponentsthatijaj=(∂x/∂x)xai.(3.14)Weproceedtoprovideimportantexamplesofdifferential1−forms.Themostpervasiveoneotherthanthedifferentialofthecoordinatesisthedifferentialofscalar-valuedfunctions,df=(∂f/∂xi)dxi=fdxi.(3.15),iNoticethenewnotationforpartialderivativesin(3.15).Severalimportant1−formsinphysicsarespecificcasesofthisequation.Butwhicharetheman-ifoldsinsuchcases?Therearemanydifferentonesinphysics.Oneofthemis,forinstance,themanifoldofstatesofathermodynamicalsystem.TypicalcoordinatepatchesonthismanifoldareconstitutedbytheenergyE,volumeV,andnumbersofmoleculesN1,N2,...ofthedifferentsubstancesthatconstitute,say,amixtureofgases(ifsuchisthesystem).AfunctiononthemanifoldistheentropySofthesystem,S=S(U,V,N1,N2,...).Thedifferentialoftheentropyfunctionisa1−formwithcoefficientsthatcontainthetemperatureT,pressureP,andchemicalpotentialsμ1,μ2,namely:−1−1−1−1dS=TdU−PTdV+μ1TdN1+μ2TdN2...(3.16)Themostfamiliarmanifold,however,isthespace-timemanifold,whichhas4dimensions.Itisdescribedbyatimecoordinateandthreespatialcoordinates.Inadditiontobeingadifferentiablemanifold,space-timehasmuchmoreaddi-tionalstructure.Itisatthelevelofthisadditionalstructure,whichwillbethesubjectoflaterchapters,thatthespace-timeofNewtonianphysicsdiffersfromthespace-timeofspecialrelativityandfromthespace-timesofEinstein’stheoryofgravity(alsocalledgeneralrelativity).Thecontentsofthischapterappliesequallywelltoallthesedifferentspace-timemanifoldsandweshallproceedtodiscusssomespecificphysicalexamplesof1−formsinspace-time.

4628CHAPTER2.DIFFERENTIALFORMSThephaseofawaveisusuallygiveninCartesiancoordinates,φ(t,x,y,z).Itsdifferentialisdφ=(∂φ/∂t)dt+(∂φ/∂x)dx+(∂φ/∂y)dy+(∂φ/∂z)dz=ωdt−kxdx−kydy−kzdz,(3.17)whereω,−kx,−ky,and−kzarenothingbutthosepartialderivativesuptosign.Asweshalleventuallysee,thereisnoroomforCartesiancoordinatesinthespace-timesofEinstein’sgravitytheory.However,theconceptofphaseofawaveisofsuchanaturethatitdoesnotrequirethespace-timetobeflat,thewavetobeaplanewaveandthecoordinatestobeCartesian.Wealwayshavedφ=(∂φ/∂x0)dx0+(∂φ/∂xi)dxi,(3.18)withsummationoveri’s.Generalizedfrequencyandwave-numberscanbede-finedask≡∂φ/∂x0andk≡−∂φ/∂xisothat0i0123dφ=k0dx−k1dx−k2dx−k3dx,(3.19)intermsofarbitrarycoordinatesofwhateverspace-timedifferentiablemanifoldisconcerned.AnotherimportantphysicalexampleisgivenbytheactionS0ofafreepar-ticlewhenconsideredasafunctionofthespace-timecoordinates.TheactionistheintegrandinthevariationalprincipleδdS=0thatgivesrisetotheequa-tionsofmotionofaphysicalsysteminclassicalphysics(Weusethesubscriptzerowhenthesystemisjustafreeparticle).Inthiscase,wehave:−dS0=−(∂S0/∂t)dt−(∂S0/∂x)dx−(∂S0/∂y)dy−(∂S0/∂z)/dz=Edt−pxdx−pydy−pzdz,(3.20)whereEistheenergyandthepi’sarethecomponentsofthemomentumoftheparticle.Asinthecaseofthephaseofawave,wecouldgeneralizethisconcepttoarbitrarycoordinatesinthespace-timesofNewtonianphysics,specialrelativity,andgravitationaltheory.Finally,if−S1istheinteractionpartoftheactionofaunit-chargeparticleinanelectromagneticfield,wehave:−dS1=−(∂S1/∂t)dt−(∂S1/∂x)dx−(∂S1/∂y)dy−(∂S1/∂z)dz=φdt−Axdx−Aydy−Azdz,(3.21)whereφ,Ax,Ax,Ay,andAzconstitutethesocalledscalarandvectorpotentials,actuallycomponentsofthefour-potential.Saidbetter,theyarethecomponentsofadifferentialform,andwecouldexpressitintermsofarbitrarycoordinates.Itisworthreflectingonthechangeofperspectivethatthelanguageofformsrepresentsrelativetootherlanguages.Takeforinstance,Eq.(3.15).Readersareprobablyfamiliarwiththeideaofconsideringthesetofpartialderivatives(∂f/∂xi)asthecomponentsofthegradientvectorfield,andtheset(dxi)as

472.3.DIFFERENTIAL1−FORMS29componentsofdxia.Itis,however,muchsimplertoconsider∂f/∂xiastheicomponentsoftheformdf,withthedxi’sconstitutingabasisof1−forms.Nottomentionthefactthatthegradientrequiresametricstructure,anddfdoesnot.Fromthisnewperspective,dfcanbereferredtoasthegradientform.Similarly,thedifferentialofthephaseofawavecouldbereferredtoasthefrequency-wave-numberform.ThenegativedifferentialoftheactionsS0,S1,andS0+S1couldbereferredtoas,respectively,thefreeparticleenergy-momentumform,thefour-potentialform,andthegeneralizedenergy-momentumformforaparticleinanelectromagneticfield.Alltheseconceptsaremean-ingfulabinitioinanycoordinatesystem.Finally,readersshouldnoticethat(dx2+dy2+dz2)1/2,i.e.thelineelement,isneitheradifferential1−formnoradifferential2−form(theseareconsideredinthenextsection).Weusedequations(3.12)toeffectachangeofcomponentsunderachangeofbasisof1−forms.Weshallalsobeinterestedinbasesofamoregeneraltype.Twoindependentdifferentialsdx1anddx2constituteabasisof1−formsintheplane,wheretheyconstituteacoordinate(alsocalledholonomic)basis.Itisobviousthat,say,ω1=dx1,ω2=x1dx2,(3.22)alsoconstituteabasisofdifferential1−forms,sincethetransformationisinvert-ible.However,thereisnocoordinatesystem(y1,y2)suchthatω1=dy1andω2=dy2,whichisthereasonwhyitiscalledananholonomicornon-holonomicbasis.Weshallrepresentchangesofbasesasωj=Aj(x)ωi,ωj=Aji,(3.23)ii(x)ωwherexrepresentsasmanyasallthecoordinatesinthesystem,fouroftheminspacetime.Ofcourse,if(ωj)=(dxj)and(ωi)=(dxi),thenAj=∂xj/∂xiiandAjj/∂xi.i=∂xWemaywishtoexpressthedifferentialformdfintermsoftheωi’s.Weshallusethesymbolsf,iandf/itodenotethecoefficientsofdfintherespectivebasesdxiandωi,i.e.,df=fdxi=fωj.(3.24),i/jWewanttohavetheexplicitrelationbetweenf/iandf,i.Equations(3.23)applytoanypairofbasesof1−formsand,inparticular,theyapplytothepairofbases{dxi}and{ωi}.Thus,defining{ωi}≡{ωi},wehavejjjiω≡ω=Aidx,(3.25)iijijdx=Ajω=Ajω.(3.26)Substitutionof(3.26)in(3.24)yields:ijjf,iAjω=f/jω.(3.27)Hence,since(ωj)constitutesabasis,wehave,f=Ai/jjf,i,(3.28)

4830CHAPTER2.DIFFERENTIALFORMSwhichisthesoughtrelation.Expertreaderswillhavenoticedthatwhatwehavecalledandshallbecallingdifferentialformsarecalledcochainsinsomefieldsofmathematics.Sincethetermsexteriorcalculusanddifferentialformsalmostalwayscometogether,andsincesaidcalculuscertainlyappliestocochainsevenmoredirectlythantoskew-symmetricmultilinearfunctionsofvectors(thinkofwhatthegeneralizedStokestheoremstates),itmakessensetousethetermdifferentialformtorefertowhatgoesbythenameofcochains,atleastwhendoingcalculus.2.4Differentialr−formsWeproceedtointroducedifferentialformsof“highergrade”.Youhavemettheminthevectorcalculus,wheretheyaredealtwithinanunfortunateway.Indeed,considertheintegralp2dpdq(4.1)andperformthechangeofvariablest=(p+q)/2,x=(p−q)/2or,equivalently,p=t+x,q=t−x.Wereadilygetthatdpdq=(dt+dx)(dt−dx)=dt2−dx2.Sotheexpressionp2dpdqisincontrovertibly,222(t+x)(dt−dx).(4.2)However,theexpression(t+x)2(dt2−dx2)(4.3)doesnotmakesenseand,inanycase,itisnotthewayinwhichwechangevariablesinamultipleintegral.Onemaywonderwhattheproblemiswithwhatwehavedone.Theproblemistheinadequatenotationdpdq,aswenowexplain.Whenwewritedpdqundertheintegralsign,wereallymeana“new”typeofproduct,whosepropertiesweproceedtodiscuss.Wecangivetwoorientationstosurfaceelementsbygivingtwodifferentorientationstotheboundary.Thesetwoorientedelementsofsurfaceintegralcanberepresentedrespectivelybydp∧dq(read“dpexteriorproductdq”)anddq∧dp(read“dqexteriorproductdp”,andalso“dqwedgedp”).Theskew-symmetryremindsusofthecrossproduct,whichactuallyisthecompositionoftheexteriorproduct,whichweareabouttodefine,withtheassignmentofasetofquantitieswithoneindextoasetofquantitieswithtwoindices.Theinterpretationoftheexteriorproductasanelementofsurfaceintegralrequiresthat,aspartofthedefinitionofthatproduct,wemusthavedp∧dq=−dq∧dp.(4.4)Consequentlydp∧dp=dq∧dq=0,(4.5)

492.4.DIFFERENTIALR−FORMS31whichcorrespondstothelackofintegrandsproportionaltodp2anddq2.Supposenowthatthetwoelementsoflineintegralconstitutingthesurfaceelementweregivenbythe1−formsα=a1dp+a2dqandβ=b1dp+b2dq.Thecorrespondingelementofsurfaceintegralwouldbewrittenasα∧β=(a1dp+a2dq)∧(b1dp+b2dq).(4.6a)Weshouldliketoreducethistothebasicobjectdp∧dq.Onereadilyinfersthatlinearitywithrespecttobothfactorsshouldbeassumed(assuggestedforinstancebythecase(a2=b1=0)).Therefore:α∧β=a1b2dp∧dq+a2b1dq∧dp=(a1b2−a2b1)dp∧dq.(4.6b)Ifαandβarethedifferentialformsdtanddxintermsofdpanddq,then(a1b2−a2b1)issimplytheJacobian∂(t,x).(4.7)∂(p,q)Thus:∂(t,x)dt∧dx=dp∧dq.(4.8a)∂(p,q)Ifweinterchangetheroleofthevariablesinthisexpression,weget∂(p,q)dp∧dq=dt∧dx.(4.8b)∂(t,x)Substitutionof(4.8b)in(4.8a)yieldstheknownfactthatthesetwoJacobiansaretheinverseofeachother.Thisispreciselythewayinwhichwechangevariablesinadoubleintegral.Ingeneralandnotonlyintwodimensions,wecalldifferential2−formstheskew-symmetricproductsoftwodifferential1−forms,andsumsthereof.Letusreturnto(4.1).Assumethatinsteadofp2dpdqweconsideredp2dp∧dq.Thendp∧dq=(dt+dx)∧(dt−dx)=−dt∧dx+dx∧dt=−2dt∧dx.(4.9)Thesurfaceintegralwouldbecome−2(t+x)2dt∧dx.(4.10)Thecoefficient−2isthe(constant)Jacobianofthetransformation.Theminussignexpressesthefactthattheorientationofthe“setsofaxes”(t,x)and(p,q)isthereverseofeachother.Weconcludethatweobtainedtheinappropriateexpressiondt2−dx2in(4.3)becauseinappropriatenotationledustochangevariablesindpdqratherthanindp∧dq.Letusforgetdifferentialformsforthemomentandrecallthecrossproduct,a×b.Itisdesignedasavectorthathasthesamedirectionasoneofthetwo

5032CHAPTER2.DIFFERENTIALFORMSorientednormalstotheparallelogramwithsidesaandb.Themagnitudeofa×bistheareaoftheparallelogram.Wecouldthinkofa∧basrepresentingtheparallelogramitself.Hence,inafiststepweassigntothepair(a,b)theparallelograma∧band,inasecondstep,thenormaltotheparallelograma×b.Withdifferentialforms,oneignorestakingthenormalandstopsatthewedgeproduct.Weshallreachtheconceptofdifferentialr−formbymeansoftheso-calledexteriorproduct∧ofanumberrofdifferential1−forms.Thisisdefinedasanalgebraicoperationthatsatisfiestheproperties(a)itislinearwithrespecttoeachfactorand(b)itisanticommutative(i.e.skew-symmetric)withrespecttotheinter-changeofanytwofactors.Theresultoftheexteriorproductofanumberrofdifferential1−forms(ω1,...,ωr)iscalledasimpler−formandisdenotedasω1∧...∧ωr.Anylinearcombinationofr−formsalsoiscalledanr−form,whichingeneral,neednotbesimple.Forinstance,if(t,x,y,z)arefourindependentcoordinates,the2−formdt∧dx+dy∧dzisnotsimple.If(x,y,z)arethreeindependentcoordinates,theformdx∧dy+dx∧dz+dy∧dzcanbewrittenasasimple2−forminmanydifferentways.Anexchangeoftwo1−formfactorsinaproductchangesthesignoftheproduct.Toseethismovethelastofthetwofactorstobeexchangedtojustbehindtheotherone,andthenthisonetowherethelastonepreviouslywas.Ifsisthenumberofplacesinvolvedinthefirstmove,thenthenumberinvolvedinthesecondmoveiss+1foratotalof2s+1,whichisanoddnumberandthusimpliesachangeofsign.Becauseofskew-symmetry,asimpler−formbecomeszeroiftwoofthefactorsintheproductcoincide.Buttheinterchangeofapairofindicesinanon-simpler−formdoesnotnecessarilychangeitssign.So,theinterchangeofthesuperscripts3and4inω1∧ω2+ω3∧ω4yieldsω1∧ω2+ω4∧ω3whichisequaltoω1∧ω2−ω3∧ω4.Considerallpossibleformsthatwecanconstructbyexteriorproductofthedifferentials(dx1,dx2,dx3)ofthreeindependentcoordinates(x1,x2,x3).Thereareonlysixnon-nullproductsofthesethreeone-forms.Theycanbewrittenasdxi1∧dxi2∧dxi3,(4.11)where(i1,i2,i3)is(1,2,3)orapermutationthereof.Obviouslydxi1∧dxi2∧dxi3=εdx1∧dx2∧dx3,(4.12)i1i2i3whereεi1i2i3isthesignofthepermutation(i1,i2,i3).Itfollowsthatall3−formsin3-dimensionalmanifoldsareascalarfunctiontimesdx1∧dx2∧dx3.Thisappliestoanycoordinatesystem.Tobespecific,let(x,y,z)and(r,θ,φ)respec-tivelydenotetheCartesianandsphericalcoordinatesofourusual3-space.It

512.4.DIFFERENTIALR−FORMS33followsfromtheaboveconsiderationsthatthe3−formdx∧dy∧dzisamultipleofthe3−formdr∧dθ∧dφ.Thusdx∧dy∧dz=f(r,θ,φ)dr∧dθ∧dφ.(4.13)Weshallsoonfindthefunctionf(r,θ,φ).Letusnowcommentonthepostulateoflinearity.Itimpliesthat,ifoneofthefactorsisalinearcombinationofone-forms,thegivensimpler−formcanbewrittenasasumofothersimpler−forms.Asanexample,ifω1=α+βthenω1∧...∧ωr=α∧ω2∧...∧ωr+β∧ω2∧...∧ωr.(4.14)Wearenowreadytorelatedr∧dθ∧dφtodx∧dy∧dz.TheCartesiancoordinatesaregivenintermsofthesphericalcoordinatesbytherelationsx=rsinθcosφ,y=rsinθsinφ,z=rcosθ.(4.15)Differentiationoftheseequationsandsubstitutionindx∧dy∧dzyieldsdx∧dy∧dz=(sinθcosφdr+rcosθcosφdθ−rsinθsinφdφ)∧(sinθsinφdr+rcosθsinφdθ+rsinθcosφdφ)∧(cosθdr−rsindθ).(4.16)Ifwedenotetheeightdifferentone-formsintherighthandsideofthisequationbythesymbolsA1toA8inthesameorderinwhichtheyhaveappeared,wegetdx∧dy∧dz=A1∧A6∧A8+A2∧A6∧A7+A3∧A4∧A8+A3∧A5∧A7(4.17)=−r2sin3θcos2φdr∧dφ∧dθ+r2sinθcos2θcos2φdθ∧dφ∧dr+r2sin3θsin2φdφ∧dr∧dθ−r2cos2θsinθsin2φdφ∧dθ∧dr=r2sinθdr∧dθ∧dφ.Anefficientwayofgettingthetermsinvolvedistostartbyconsideringthatthereisnodφfactorinthelastlineof(4.16).Theprecedingexerciseembodiesseveralmanipulationswithdifferentialr−forms,andwerecommendusingitforpractice.Givennlinearlyindependent1−forms,thereareasmanylinearlyinde-pendentr−formsastherearer-combinationsofnelements,i.e.astherearesubsetswithrobjectstakenfromagivensetofndifferentobjects.Ifthespaceofthe1−formsisndimensional,ther−formsconstituteaspaceofdimensionn!/r!(n−r)!.Inparticular,thespaceofthen−formsisofdimension1.Alltheformsofgrader>narenull.Asanexample,considerthespaceof2−formsina3-dimensionalmanifold.Abasisinthisspaceisgivenby(dx1∧dx2,dx2∧dx3,dx3∧dx1).However,wemightaswellhavetaken(ω1∧ω2,ω2∧ω3,ω3∧ω1)asabasisof2−formsif(ω1,ω2,ω3)representsabasisof1−forms.

5234CHAPTER2.DIFFERENTIALFORMSInEinstein’stheoryofgravity,oneoftencomesacrosschangesofbasesofr-forms,especially2−forms.Acoupleofexampleswillhelpustofurtherbecomefamiliarwiththeseideas.Ausefulbasisof1−formsincentralbodyproblemsinEinstein’stheoryofgravityisgivenby:ω0=eλ(r)dt,ω1=eμ(r)dr,ω2=rdθ,ω3=rsinθdφ,(4.18a)whereνandμarefunctionsofthevariabler.Letusdenote(dt,dr,dθ,dφ)as(dx0,dx1,dx2,dx3).Weareinterestedinwritingthe2−formsdxi∧dxjaslinearcombinationsofthe2−formsωk∧ωi.Inthiswaytheexpressionintermsofdxi∧dxjcanbetranslatedintoanexpressionintermsofωk∧ωi.Fromthegivenequations,weobtain:dt=e−λω0,dr=e−μω1,dθ=r−1ω2,dφ=r−1(sinθ)−1ω3.(4.18b)Thus,forexample,dr∧dt=e−(λ+μ)ω0∧ω1,dr∧dθ=r−1e−μω1∧ω2(4.19)−1−1−μ13−2−123dr∧dφ=r(sinθ)eω∧ω,dθ∧dφ=r(sinθ)ω∧ω.Supposenowthatwearegiventhefollowing,slightlymorecomplicatedproblem,wheretheforms(ωi)arenotsimplyproportionaltotheforms(dxi)butarerathergivenby2222021−αr/22−αr/23ω=dt−ardφ,ω=edr,ω=edz,ω=rdφ,(4.20a)whereαandaareconstants.WeshallconsidertheseformsagaininChapter10.Weget2222dt=ω0+arω3,dr=eαr/2ω1,dz=eαr/2ω2,dφ=r−1ω3.(4.20b)Itthenfollowsthat2222dt∧dr=eαr/2ω0∧ω1−areαr/2ω1∧ω3,2222dt∧dz=eαr/2ω0∧ω2−areαr/2ω2∧ω3,22(4.21)dt∧dφ=r−1ω0∧ω3,dr∧dz=eαrω1∧ω2,2222dr∧dφ=r−1eαr/2ω1∧ω3,dz∧dφ=r−1eαr/2ω2∧ω3.Ascanbeseen,thereisnothingcomplicatedaboutexteriorproductsof1−forms.Letusnowconsiderthecasewhenthegradesofthefactorsaregreaterthanone.2.5ExteriorproductsofdifferentialformsTheconceptofexteriorproductsof1−formscanbeextendedtor−forms,the“grades”ofthefactorsnotbeingnecessarilythesame,muchlessone.Considertheproductoftwoforms,ρandσ,ofrespectivegradesrands,ρ=ωi1∧...∧ωir,σ=ωj1∧...∧ωjs.(5.1)

532.6.CHANGEOFBASISOFDIFFERENTIALFORMS35Theirexteriorproductisdefinedas:ρ∧σ=ωi1∧...∧ωir∧ωj1∧...∧ωjs.(5.2)Noticethatwedidnotneedtoconsideracoefficientinρ(andinσ),sayρ=Rωi1∧...∧ωir,sincethecoefficientcanalwaysbeabsorbedinoneofthefactors,e.g.ρ=(Rωi1)∧...∧ωir.Theexteriorproductislinearwithrespecttoeachfactor,andisassociativewithrespecttothemultiplicationbyascalarf(0-gradeform),i.e.(fρ)∧σ=ρ∧(fσ)=f(ρ∧σ).(5.3)Thefollowingpropertyisveryuseful:(ρ∧σ)=(−1)rsωj1∧...∧ωjs∧ωi1∧...∧ωir=(−1)rsσ∧ρ.(5.4)Thiscanbeseenasfollows.In(5.2),wemoveωj1tothefrontbyexchangingplaceswithωir,thenwithωir−1andsoonuntilitexchangesplaceswithωi1.Indoingso,therehavebeenrchangesofsign.Werepeattheprocesswithalltheωj’s.Thenumberofchangesofsignisrs,andtheaboveexpressionfollows.Wefurtherdemandthedistributiveproperty,(ρ1+ρ2)∧σ=ρ1∧σ+ρ2∧σ,(5.5)whereρ1andρ2areusuallyofthesamegradeintheexteriorcalculus,butneednotbeso.Similarlyρ∧(σ1+σ2)=ρ∧σ1+ρ∧σ2.(5.6)Exteriorproductsofdifferentialformswillbeusedextensivelyintheexteriorcalculus,alsocalledCartancalculus,whichwillbepresentedinchapterfour.2.6ChangeofbasisofdifferentialformsDifferentialformsareinvariants:theydonottransform,theircomponentsdo.Achangeofbasisofdifferential1−forms,likefromthebasis(dp,dq)to(dx,dy)in2-D,inducesthechangeofbasisintheone-dimensionalmoduleofdifferential2−forms,fromdp∧dqtodx∧dy.Consider3−formsona3-Dmanifold.Theyalsoconstitutea1-dimensionalspace.Thusdx1∧dx2∧dx3constitutesabasis.Ifthedxi(i=1,2,3)arethedifferentialsofasetofthreeindependentcoordinates,achangeofcoordinatesfromx’stoy’sinducesachangetothebasisdy1∧dy2∧dy3.Ifdyi=Aidxj,(6.1)jthecorrespondingchangeofbasisof3−formsisgivenbydy1∧dy2∧dy3=A1A2A3dxi∧dxj∧dxk.(6.2)ijk

5436CHAPTER2.DIFFERENTIALFORMSWeendupontherightsidewiththefullpanoplyofallpossibleexteriorproductsofthethree1−formsdxi,butallofthemarejustproportionaltoeachother.WeshalldistinguishasumsuchasAdxi∧dxj∧dxk,(6.3)ijkwhichisextendedtoallpossiblevaluesoftheindices,fromasumlikeA(dxi∧dxj∧dxk),(6.4)ijkwheretheparenthesisisusedtodenotesummationonlyoverallpossiblecom-binationsofindiceswithirthethreeindicesmaybeany.Itshouldbeclearalsothat,forr−forms,wehavei1i2ir1i1i2irAi1i2...ir(dx∧dx∧...∧dx)=Ai1i2...irdx∧dx∧...∧dx,(6.7)r!withtheAtotallyskew-symmetric.i1i2...ir

552.7.DIFFERENTIALFORMSANDMEASUREMENT37Letuspracticewiththesimpleexampleoftheelectromagnetic2−form.Itsversionofthe(6.4)typeisF=Exdt∧dx+Eydt∧dy+Ezdt∧dz−Bzdx∧dy−Bydz∧dx−Bxdy∧dz.(6.8a)Ifwewishtowriteitintermsofthe2−formsgeneratedbydx,dy,dzanddtwherex=γ(x−vt),y=y,z=z,t=γ(t−vx),(6.9a)weusethetransformationsinversetothese,whicharegivenbyx=γ(x+vt),y=y,z=z,t=γ(t+vx).(6.9b)Wedifferentiate(6.9b)atconstantv,substitutein(6.8a)andobtainF=Edt∧dx+γ(E−vB)dt∧dy+γ(E+vB)dt∧dzxyzzy(6.8b)−γ(Bz−vEy)dx∧dy−γ(By+vEz)dz∧dx−Bxdy∧dz.IfwedenoteasE,E,E,−B,−Band−BthecoefficientsofFwithxyzxyzrespecttotheprimedbasisof2−forms,weobtain(E,B)componentsintermsof(E,B)components.Ontheotherhand,wemightbegivenFintheform1ijFijdx∧dx,(6.10)2!withFijskew-symmetricandwithcomponentswhicharenothingbutthecoef-ficientsin(6.8a).WethenhaveijFkm=Ak¯Am¯Fij.(6.11)wheretheAiarereadfromtheequations(6.9b),withxi=(t,x,y,z).k¯2.7DifferentialformsandmeasurementInthissection,wediscusslengths,areasandvolumes.Forsimplicity,letusstartwiththe3-dimensionalmanifoldofdailylife(Euclidean3-space,tobedefinedlater).Weaskourselveswhethervolumesaregivenbytheintegral2dx∧dy∧dz=rsinθdr∧dθ∧dφ(7.1)overcorrespondingdomainsorbytheintegraldr∧dθ∧dφ.(7.2)Thesolutioniswell-known,morebecauseofone’sfamiliaritywithEuclideanspacethanbecauseoffamiliaritywiththetheoryofintegration.But,what

5638CHAPTER2.DIFFERENTIALFORMShappensinthecasewhenthen−dimensionalmanifold(say,then-sphere,nottoconfusewiththen-ball),unlikeEuclideanspace,doesnotadmitnCartesiancoordinates?Inexperiencedreadersshouldnotoverlookthefactthat,whenweuseCarte-siancoordinates(x,y,z)onthe2-sphere,thenumberofsuchcoordinatesisn+1,withn=2;theyarenotindependentinthespheresince,giventhexandyofapointonit,itszcoordinateisdetermined.Letusconsideran-manifoldgiveninsuchawaythatwedonotknowwhetheritiscontained(althoughitis)insomeEuclideanspaceofhigherdimension,andwhichthusadmitsCarte-siancoordinates.Wewouldnotknowwhethertoreadthen-volumedifferentialformasthecoefficientofthen−formdx1∧...∧dxnassociatedwithagivencoordinatesystemorwiththeformdy1∧...∧dynassociatedwithsomeothercoordinatesystem.Lookingforguidance,wecanrefertothetransformationofn−forms.Achangeofbasesof1−formsinducesachangeofbasesofn−formsgivenbyω1∧...∧ωn=A1A2...Anωi1∧ωi2∧...∧ωini1i2in=|Aj1∧ω2∧...∧ωn,(7.3)i|ωjjwhere|Ai|isthedeterminantofthematrixwhoseelementsaretheAi’s.If(ωi)=(dyi)and(ωi)=(dxi),Eq.(7.3)becomesdy1∧dy2∧...∧dyn=J(∂y/∂x)dx1∧dx2∧...∧dxn,(7.4)whereJ(∂y/∂x)istheJacobian.Butwehaveaswelldx1∧dx2∧...∧dxn=J(∂x/∂y)dy1∧dy2∧...∧dyn.(7.5)Fromtheseexpressions,thereisnowaytotellwhich3−form,whetherdx1∧dx2∧...∧dxnordy1∧dy2∧...∧dynornoneofthetworepresentsthevolumeelement.Actually,onesuchelementmaynotbedefined.Whatwearemissingisayardstickforeverydirectionsothat,evenalongacoordinateline,weshallbeabletotellhowlongsomethingis.Surfacesandvolumesmakeclearthatwealsoneedtoknowhowmuchacoordinatelinex1istiltedwithrespecttoanothercoordinatelinexj(i.e.,the“angleoftwodirections”).Allthisisachievedbytheintroductionofametric,whichwillbedoneinalaterchapter(angle,likelength,isametricconcept).Weconcludethat,unlessweintroducemorestructureonourmanifold,wecannotsaywhattheareaofasurfaceis.Wecan,however,calculatefluxesthroughsurfacesand,ingeneral,wecanintegrate.Asanexample,thefluxofthemagneticfieldthroughasurfaceatconstanttimeisgivenbytheintegralofthenegativeoftheelectromagnetic2−formafterremovingthetermscontainingdtasafactor.2.8DifferentiablemanifoldsDEFINEDWenowgiveaformaldefinitionofdifferentiablemanifoldM,previouslytreatedinformally.AdifferentiablemanifoldofdimensionnandclassCrconsistsofa

572.8.DIFFERENTIABLEMANIFOLDSDEFINED39setMof“points”andasetUM(calledatlas)ofcoordinatepatchesorcharts.Achartconsistsofapair(U,fU)whereUisasubsetofMandfUisaone-to-onecorrespondencefromUontoanopensubsetf(U)ofRn.OnereferstoUnasthedimensionofthemanifold.Letthesymbol“◦”denotecompositionoffunctions.Themanifoldhastosatisfythefollowingtwoconditions:1)ThesetsU⊂UMcoverthewholesetM.2)IfUandUoverlap,f◦f−1isafunctionfromf(U∩U)intoRn12U2U1U112thatiscontinuoustogetherwithallitsderivativesuptoorderr,i.e.itisCr.Letusnowdescribeinlesstechnicalwordstheobjectthatwehavejustde-fined.Opensetsarerequiredsoastoavoidboundaries,sincepointsinbound-ariesdonothaveneighborhoodsthatlooklikeRn.Theone-to-oneandontorequirementsforthechartstogethermakethefunctionsfUinvertible.NoticethatfUi(Ui)isnothingbutasetofcoordinatefunctions.IfP∈U,fU(P)rep-resentsaset(xi(P))ofcoordinatesofthepointP.Wehaveguaranteedthatwecanintroducecoordinatesalloverthesetofpoints(firstconditionabove).Asetofcoordinatesneednotcoverthewholemanifold.Forinstance,thereisnoonechartthatcoversthewholesphericalsurface(i.e.withjusttwobutneverlessthantwocoordinates).Thusthevaluesθ=0andθ=πdescribethepolesofthesphericalcoordinatesystemregardlessofthevalueofφ.Werequirethat,piecebypiece,wecoverthewholesetofpointsMwithcharts,alsocalledcoordinatepatches.Condition2achievesthatwhentwochartsoverlap(theoverlapbeingU1∩U),thetransformationfromonesystemofcoordinatestotheotherisCr.Two2charts(U,f)and(U,f)thatsatisfycondition2abovearesaidtobeCr1U12U2compatible.Onceanatlashasbeengivenorfound,weshallincludeintheatlasallthechartsthatareCrcompatiblewiththosealreadythere.Theatlassocompletediscalledamaximalatlas.Theatlasinthedefinitionofmanifoldwillbeunderstoodtobemaximal.LetIbetheidentitymaponRn;Rnthenbecomesamanifoldifweaddtoittheatlasconstitutedbytheonechart(Rn,I)togetherwithallthechartsthatareCrcompatiblewithit.Asasummary,aCrdifferentiablemanifoldisthepairmadeofasetofpointsandamaximalatlasthatcoversthesetofpointsofthemanifold.Changesofchartwithintheatlas(i.e.coordinatetransformations)arerequiredtobeCrfunctions.Inthefollowing,weshallspeakindistinctlyofMandM.Letusnextconsiderdifferentiablefunctionsonmanifolds.Letgbeafunc-tionfrommanifoldM1tomanifoldM2andletU1⊂UM1andU2⊂UM2.−1WesaythatgisdifferentiabletoorderratapointPofM1iffU2◦g◦fU1isdifferentiabletoorderratfU(P).Becauseofcondition2onmanifolds,thisdefinitiondoesnotdependonwhichpatcheswechoosearoundP.Thefunctiongmayinparticularbearealvaluedfunction,i.e.M2maybeRitself.Inthiscase,fU2issimplytheidentitymap.

5840CHAPTER2.DIFFERENTIALFORMSSincethecompositemapg◦f−1takesusfromRntoR1(seeFigure1),weUdefinethedifferentialdgofgasdg=(∂g/∂xi)dxi,where∂g/∂xi=∂(g◦f−1)/∂xi,(8.1)Uwithxidefinedby(xi)=f(P).ItisconvenienttointroducethenotationUnRgMfU1R-1iU()gfU/x1RFigure1:Manifoldrelatedconceptsg(xi)forg◦f−1.TheexpressionfordggivenabovethenbecomesthedgoftheUcalculusinRn.Mostauthorswithautilitarianapproachtomathematicsusuallyeliminateanyreferencetotheclassofthemanifold.Explicitlyorimplicitly,theyassume,likeweshalldo,thattheclassisashighasneededforourpurposesatanygiventime.Tothosewhodonotthinklikeexpertsonanalysis(whichincludesthisauthor),letussaythatthereare1-Dfigureswhicharenot1-Dmanifoldsinthestrictsensejustdefined.Acontinuouscurveisa1-Ddifferentiablemanifoldifitdoesnotintersectitself.Pointsofintersectionorpointsoftangency(thinkoftheletterq)arenotdifferentiablemanifolds.Theyfailtobesobyvirtueoftheintersectionand/ortangencypoints.Asforpolygons,theyarenotdifferentiablemanifoldsbecauseofnotexistenceofaderivativeatthevertices(onehastofirstsmooththeminordertoapplyanyspecificresults).Weshallassumethevalidityintheonlycasewhereitisneededinthisbook(inalemmatotheGauss-BonnettheoreminAppendixA).2.9AnotherdefinitionofdifferentiableMANIFOLDTheconceptofdifferentiablemanifoldjustgivenistheusualoneinthemodernliterature,butitisnottheonlyacceptedone.Wenowpresentanequivalentone[92](notcompletely,sinceitonlyhastodowithEuclideanstructure)inwhichonedoesnotmakeexplicitreferencetothecharts.Letstartbysayinginplainlanguagewhatwegotintheprevioussection.Chartsplaytheroleofprovidingthearenathatsupportsdifferentiationinacontinuoussetofpoints.Thepurposeofhavingdifferentchartsandnotjustoneistobeabletoadequatelycoverallpointsintheset,onesmallpieceatthe

592.9.ANOTHERDEFINITIONOFDIFFERENTIABLEMANIFOLD41time.Wethenendowthesetwithadditionalstructure,materializedthroughasystemofconnectionequationsandasystemofequationsofstructure,thelatterspeakingofwhethertheformerisintegrableornot.Forsimplicity,consider2-Dsurfacesandtrytoignorethe3-DEuclideanspaceinwhichwevisualizethem.Youmaythinkthatyouareignoringthe3-Dspace,butyouarenotwhenyoudealwiththenormaltothesurface:itisnotdefinedexceptthroughtheembeddingofthesurfaceinahigherEuclideanspace.Thus,withoutthe(inthepresentcasethreedimensional)Euclideanspace,onehastothinkofevena2-Dsurfaceinjustsomeabstract,mathe-maticalway.ButtheequationsofstructuredenyingeneralthatthesetisaflatEuclideanstructure,unlesswearespeakingofplanes,conesandcylinders(i.e.surfacesdevelopableonaplane).Themoststructurewethengetonthosemanifoldsisthatsmallneighborhoodsofeachpointarealmostlikesmallpiecesof2-DEuclideanspace.Hence,ithelpstothinkofanalternativeconceptofdifferentiablemanifold.Considerthepracticalaspectsofdefiningasetofwhichwesayitisadifferen-tiablemanifold.A2-sphereisadifferentiablemanifold,butsoisthepuncturedsphere,andalsotheidealizedpartofsurfaceoftheearthcoveredbyland,andalsothesmoothedpartcoveredbysea,andthealsoidealizedsurfaceofanycon-tiguouscountry,etc.Whatisbehindthepossibilityofdistinguishingbetweenallthesecases?Theyhavesomethingincommonbutarestillgivenbydifferentsubsetsofthe3-DEuclideanspace.Thisbringsustothefollowingalternativedefinition.LetEndenoteEuclideanspaceofdimensionn.AdifferentiablemanifoldisasubsetofEndefinedneareachpointbyexpressingsomeofthecoordinatesintermsoftheothersbydifferentiablefunctions”[92].Thisreplacesthecharts,butintroducesthedifferentiablefunctionstodeterminethesubsetofinterest.Whenonestartsbydefiningdifferentiablemanifoldsasintheprevioussection,thisnewdefinitionistheresultofatheorembyWhitney[92].ThetheoremassignstoeachEuclideandifferentiablemanifolddefinedasintheprevioussec-tionaEuclideanspaceofhigherdimensioninwhichitfits.Notice,inordertoavoidwronginferences,thatcommonspirals(forexample)donotfitintwodimensions,butfitinthree.TheyarebornabinitioinE3.Justn+1maynotsuffice.Thenewdefinitionisveryinteresting,butthatisnotthewayinwhich,forpracticalreasons,onedoesdifferentialgeometry.Itinvolves,inadditiontothedifferentiablemanifoldsbeingdefined,Euclideanspacesofhigherdimensions,nottospeakofthefactthatitislimitedtoRiemannianstructures,whichareofjustaveryspecifictype.(AffineismoregeneralthanEuclidean;RiemannianisEuclideanattheleveloftheverysmall).Nordoesitcompriseprojective,conformal,etc.manifolds.Wethuskeepthedefinitionoftheprevioussection,thoughonebarelyusesthatdefinitioninactualpractice,notinthisbook,notinotherbooks.Weassumethatsuchastructurejustifiesthedevelopmentofthetheoryinpartfourofthisbook.Thepresentsectionprovidesthevisionthatthetangiblethinginsuchatheoryaretheflatspaces(projective,conformal,affine,Euclidean,

6042CHAPTER2.DIFFERENTIALFORMSetc.)and(ingeneralnottotallyintegrable)associatedsystemsofdifferentialequations.Onecanhardlysaythatthenon-flatspacesareverytangible,veryimaginable.Theflatonesarepredominantlyalgebraicconcepts,andthusfarclosertowhatwecanimagine.

61Chapter3VECTORSPACESANDTENSORPRODUCTS3.1INTRODUCTIONInthisbook,thetensorcalculusisabsent,buttheconceptoftensorproductisneeded.Thefamiliarexpression(1.3)ofchapter2livesinaspacewhichisthetensorproductofaspaceofvectorsandaspaceofdifferential1−forms.Onedoesnotneedtoknowtheformaldefinitionofsuchproductsinordertobeabletooperatewiththoseexpressions.Moresubtleisthetensorproductofdifferentcopiesofoneandthesamevectorspace.Wedevotethelastsectionofthischaptertothisconcept.Wecallattentiontothefactthatwhatoneactuallydefinesisthetensorproductofspaces,tensorsbeingtheelementsoftheresultingproductstructure.Itisthetensorialcompositionlawinthestructurethatmakestensorsbetensors.Thus,elementsofastructuremaylookliketensorsbutnotbesobecauseitstensorproductbyasimilarobjectdoesnotyieldanothersimilarone.Example:thetensorproductoftwosocalledskew-symmetric(alsocalledantisymmetric)tensorsisnotskew-symmetricingeneral.Theskew-symmetricproductinthiscaseistheexteriorproduct.Theiralgebraisexterioralgebraanditselementsshouldbecalledmultivectors.Inthetensoralgebraconstructeduponsomevectorspace(oroversomemodule),thevectors(ortheelementsofthemodule)areconsideredastensorsofgradeorrankone.Thevectorsthatonefindsinthevectorcalculusarecalledtangentvectors.Theyconstitutesocalledtangentvectorspaces(oneateachpointofthemanifold),thoughitisdifficulttorealizeinEuclideanspaceswhyshouldonecallthemtangent.Thiswilllaterbecomeevident.Elementsofthespacethatresultsfromtensorproductofcopiesofatangentvectorspacearecalledtangenttensors.Weshallalsohaveanothertypeofvectorspace,calledthedualvectorspace,tobeintroducedinsection3.Indifferentialgeometry,itsmembersarecalled43

6244CHAPTER3.VECTORSPACESANDTENSORPRODUCTScotangentvectors.Onecanalsoconstructtensorproductsofcopiesofthisvectorspace.Themembersoftheseproductspacesarecalledcotangenttensors.Forthepurposeofclarity,weshallusethetermtensorinthissectiononlytorefertothoseofgradetwoorhigher.Ourtensor-valueddifferentialformsaregoingtobetensorproductsofthree“spaces”:(a)aspaceoftangentvectors(rarelyifevertangenttensors);(b)aspaceofcotangentvectors(inafewexceptionalcases,cotangenttensors)and(c)aspaceofdifferentialforms.Veryseldomshallwehavetensorsthatareaproductofcopiesofthesamevectorspace.Theterm(p,q)-valueddifferentialformwillbeusedfora(sumof)membersoftheproductofatangenttensorofgradep,acotangenttensorofgradeq(whetherskew-symmetricornot)andadifferentialform.Weshallsaythatthevaluednessis(p,q).Inviewofwhatwehavejustsaid,ourvaluednesswillbealmostexclusivelyofthetypes(0,0),(1,0),(0,1)and(1,1).Wesaid,almostexclusively.Thereareexceptions.Ontheonehand,thereisthesocalledmetrictensorandthecurvaturetensor.Thelatter—ofminorimportancerelativetothecurvaturedifferentialform—isa(1,3)tensor-valueddifferential0−form.Itisderivedfromtheaffinecurvature(1,1)-valueddif-ferential2−formandisskew-symmetricwithrespecttoonlytwoofitsthreesubscripts.Asaconcept,itisanendinitselfandnotthegatetootherconcepts.Asforthemetric,onecanadoptamoreadvancedperspectiveofgeometrythaninthisbook,aFinslerianperspective[73].Wesayperspectivesincethegeom-etryneednotbeproperlyFinslerianbut,say,Riemannian,thoughviewedasifitwereFinslerian.Inthiscase,theroleofprovidingthelengthofcurvesistakenoverfromthemetricbyadifferential1−form,whichisafterallwhatdif-ferential1−formsare:lineintegrands.TheFinslerianstructurecorrespondingtoaspacetimemanifoldisaspace-time-velocitymanifold.Thecontentsofthischapterreflectsthoseconsiderations.Thecotangentvaluednessisinducedbythevectorvaluedness(seetheargumentonddvandcurvatureinthepreface,andlaterinsection5.7).Itthusbecomesfairtosaythatmoderndifferentialgeometryisthecalculusofvector-valueddifferentialforms.Weshallstartwithaformalthoughnotexceedinglyabstractpresentationofvectorspacesinsection2(Amorecompactdefinitionofavectorspaceexists,butitresortsmoreheavilytoconceptsinalgebra).Thepointsofimportanceherearethatvectorsdonotneedtohavemagnitude,andthatwhatmakesavectoravectorisitsbelongingtoaparticularstructure.Section3dealswiththedualtangentspace.Thevectorspacesofsection2maybecalledplain.Iftheyhavetheadditionalstructureconferredbyadotproduct,theyarecalledEuclideanvectorspaces.Theyareconsideredinsection4.Insection5,wedealingreaterdetailwiththeentitiesthatconstitutethetensorproductsofdifferentstructuresthatappearinthispresentationofdifferentialgeometry.Section6,addressestheissueoftensorproductsofcopiesofthesamevectorspace.ReaderswithaccesstotheexcellentbookbyLichnerowiczontensorcalculus[53]willfindtheredeepalgebraicinvolvementwithtensors,necessaryforagoodunderstandingofthatcalculus,butunnecessaryinthisbook.

633.2.VECTORSPACES(OVERTHEREALS)453.2Vectorspaces(overthereals)Thestructure“vectorspace”appearsrepeatedlyindifferentialgeometryandisattherootofalmostanyformaldevelopment.Wecommencewithadefinitionofthisstructure.Avectorspace(overthereals)isasetofelementsu,v,...withtwooperationsobeyingthefollowingproperties:(1)Additiona.u+v=v+uCommutativeb.u+(v+w)=(u+v)+wAssociativec.0,u+0=uExistenceofZero(Vector)d.u+(−u)=0OppositeVectordefined(2)Multiplicationbyascalar(realnumber:a,b,c...a)a.1u=ub.a(bu)=(ab)uAssociativec.(a+b)u=au+buDistributiveforadditionofscalarsd.a(u+v)=au+avDistributiveforadditionofvectorsTheconceptofbasispermitsustodefinethedimensionalityofavectorspace.Abasisofavectorspaceisdefinedasasetoflinearlyindependentvectorsthatisnotcontainedinalargersetoflinearlyindependentvectors.Avectorspaceisfinite-dimensionalifithasfinitebasis.Onecanshowthat,inthiscase,allbaseshavethesamenumberofelements,sayn,whereniscallednthedimensionofthevectorspaceV.Asetofrindependentvectorsthatisnotmaximal,r

6446CHAPTER3.VECTORSPACESANDTENSORPRODUCTSnGivenavectorv∈V,itscomponentsrelativetoabasis(a1,...,an)willbedenotedas(v1,...,vn)andweshall,therefore,havev=v1a+v2a+...+vna=via.(2.1)12niThisdecompositionisunique.Ifthereweretwo(ormore),bysubtractingthemwewouldobtainthatanontriviallinearcombinationofbasisvectorsiszerowhichisacontradictionwiththeassumptionthat(a1,...,an)constitutesabasis.Incaseyoumisseditpreviously,therighthandsideof(2.1)isanabbreviationknownasEinstein’ssummationconvention:wheneverthereisarepeatedindex(onceasasubscriptandonceasasuperscript),summationfromonetothelargestpossiblevalueoftheindex,n,isunderstood.Itisimportanttonoticethatthelefthandside,v,doesnothaveindicesandthattheiofvihasbeenwrittenasasuperscript(incontradistinctiontothesubscriptiofai).Weshallsoonlearnthatthedifferenceinpositionofiinviandaconveysmuchinformation.iWeproceedtoconsiderchangesofbasis.Let(a)and(a)betwodifferentiibases,whichallowsustowriteiiv=vai=vai.(2.2)Weexpressthebasis(a)intermsofthebasis(a)asijjai=Aiaj,(2.3a)wherewehaveusedEinstein’sconvention.Wecanthinkoftheapostropheinaiasifbelongingtothesubscript,i.e.asifaiwereai.Theuseofsuchprimedsubscripts(and,lateron,alsoprimedsuperscripts)hasthefollowingpurpose.jjWewanttodistinguishthecoefficientsAifromthecoefficientsAithatgiveaiintermsofa,jjai=Aiaj.(2.3b)Noticethatthecoefficientof,say,aintermsof,say,ashouldbewrittenas12A2,inordertodistinguishitfromthecoefficientA211ofa1intermsofa2.So,A2canbeviewedasAjwithi=1andj=2.Substitutionof(2.3b)in(2.3a)1iyields,afterappropriatechangeofindices,jkai=AiAjak(2.4)and,therefore,jkkkAiAj=δi=δi,(2.5)wheretheapostrophestotheindicesoftheKroneckerdeltaarenolongerneeded.WenowuseEqs.(2.3a)and(2.3b)in(2.2)andgetvi=Aivj,(2.6a)jandvi=Aij.(2.6b)jv

653.3.DUALVECTORSPACES47ItiseasytoremembertheseequationsbyobservingthatthesuperscriptontheleftisalsoasuperscriptontherightandthatEinstein’ssummationconventionhastobereflectedinthenotation.NoticethatEqs.(2.3)expresstheelementsofonebasisintermsoftheelementsintheotherbasisofvectors.Thus,whereasviandvirepresenttheithcomponentofthesamevectorvintwodifferentbases,aandaaretwodifferentvectors.Theexpressionsiiv=via=via(2.6c)iiadmitthefollowingparallelexpressionsforexplicitlychosenai:jjai=δiaj=Aiaj.(2.6d)Letusfinallyconsidertherelationofvectorbasestothegenerallineargroup.jjItisatrivialmattertoshowthatthematrices[Ai]and[Ai]thatrelatetwobaseshavedeterminantsdifferentfromzero.Conversely,if(aj)isabasisand[Ajj)givenbyi]isanymatrixsuchthatdet[Ai]=0,then(aij{ai}=[Ai]{aj}(2.7)alsoisabasis.Thesetofallsuchn×nmatricesconstitutesthe(general)lineargroupfordimensionn,groupdenotedasGL(n).Ithasapreferredmember,i.e.theunitmatrix.Thesetofallbasesdoesnothaveapreferredmember.nByexplicitlychoosingaparticularbasisinVandexpressingallotheronesintermsofit,wegeneratethesetofallmatricesinthelineargroupGL(n).Wearethusabletoputallthebasesinaone-to-onecorrespondencewiththejelementsofthelineargroup.Thesetofelements(Ai)ofeachparticularmatrixj[Ai]inthegroupconstitutesasystemofcoordinatesinthesetofallbases.ThecoordinatizationjustintroducedisofgreatimportanceintheCartancalculus.Allvectorspacesofthesamedimensionareisomorphic.3.3DualvectorspacesGivenavectorspaceVn,thesetoflinearfunctionsφ:Vn→R1constitutea∗nvectorspaceVofthesamenumbernofdimensionsifweintroduceinsaidsetalawofadditionofthosefunctionsandalawofmultiplicationbyscalars(thensamesetofscalarsasforV).Inthisbook,unlikeinmostmodernpresentationsofdifferentialgeometry,wemakeveryminoruseofthisconcept.Hence,ourtreatmentofdualspaceswillbeschematic.i∗nBases(φ)inVcanbechosensuchthatiiφ(aj)=δj.(3.1)nTheyarecalleddualbases,meaningdualofcorrespondingbasisofV.Anylineartransformationcanbewrittenintermsofabasisasiφ=λiφ,(3.2)

6648CHAPTER3.VECTORSPACESANDTENSORPRODUCTSwherethecoefficientsλ’sareitscomponents.Achangeofbasisa→ainVniiiii∗ninducesachangeofbasisφ→φinVsothatEq.3.1continuestohold,i.e.,sothatφi(a)=δi.IfAisthematrixforthechangeofbasis{a}→{a},jjiithen[AT]−1isthematrixforthechangeofbasis{φi}→{φi},wherethesuperscriptTstandsforthetransposematrix.Inmodernnotation,Eq.(3.1)isalsowrittenasiiφaj=δj.(3.3)Itreadilyfollowsthatφi(vja)=vi.(3.4)jnInlaterchapters,weshallrefertospacesVastangentvectorspaces,and∗ntospacesVascotangentvectorspaces.Thereasonforthesedenominationswillthenbecomeclear.3.4Euclideanvectorspaces3.4.1DefinitionTheconceptofEuclideanvectorspaceisattherootoftheoftenmentionedbutnotalwayswelldefinedconceptofEuclideanspace.Inthisbook,thetermEuclideanwillcomprisealsowhatisoftencalledpseudo-Euclidean(soontobedefined),exceptforspecificationtothecontrary.AEuclideanvectorspaceisnnnavectorspaceVtogetherwithamapV×V→Rcalleddotorscalarproductsuchthat:(a)u·v=v·u(b)(au)·v=u·(av)=a(u·v)(c)u·(v+w)=u·v+u·w(d)u·v=0∀u⇒v=0.Givenavectorbasis(ai),letgbethematrixwhoseelementsaregivenasgij=ai·aj=gji.(4.1)Thusu·v=uivig.(4.2)ijProperty(d)impliesthatdetg=0.Indeedifu·vequalszeroforallu,i.e.ifguivi=0,∀{ui},ijthengvi=0,∀i.ij

673.4.EUCLIDEANVECTORSPACES49Thisisahomogeneouslinearsysteminthevj’s.Itsonlysolutionisvi=0,forallj.Thendetg=0,whichwewantedtoprove.Asaparticularcaseof(4.2),wehavethat,foranyvectorv,v·v=gvivj.(4.3)ijIftheexpressiongvivisdefinitepositive,thespaceissaidtobeproperlyijjEuclidean.Ifitisdefinitenegative,onecanredefinethedotproductasthenegativeofthepreviousonesothatitbecomesdefinitepositiveandthevectorspaceisagainproperlyEuclidean.Iftheexpression(4.3)isnotdefinite,onespeaksofapseudo-Euclideanvectorspace.Thesymbolsusedmayremindreadersofthoseusedindifferentialgeometry.Butwearedealingherewithalgebra.Thegijdonotdependoncoordinateshere,astherearenotyetcoordinatestospeakof.Eventually,inlaterchapters,weshallhavefieldsofvectorspaces.gijwillthenbepointdependentandweshallstarttorefertogijasthecomponentsofthemetric.3.4.2OrthonormalbasesInproperlyEuclideanvectorspaces,abasis(ai)issaidtobeorthonormalifai·aj=δij.(4.4)“Ortho”standsforai·aj=0i=j,(4.5)and“normal”standsforai·ai=1(4.6)(withoutsummationoverrepeatedindices).Intermsofthecomponentsviofavectorvrelativetoanorthonormalbasis,wehave,i2v·v=(v).(4.7)ii21/2Thequantity(v)isdenotedasthemagnitudeornormofv.Ifavectorspaceispseudo-Euclidean,nobasiscanbefoundinitsuchthatEq.(4.4)issatisfiedforallvaluesoftheindices.Indeed,Eq.(4.4)impliesEq.(4.7)and,therefore,positivedefiniteness,contrarytothehypothesis.InsteadofEq.(4.4),onenowhasai·ai=±1,(4.8)wherethe+signappliestoatleastonebutnotallthevaluesofthesubscript.Inthefollowing,weshallusethetermorthonormalbasestorefertobothproperlyorthonormalandpseudo-orthonormalbases.OnesaysthattheorthogonalgroupO(n)actstransitivelyonthesetoforthonormalbasesinndimensions,meaningthatanyorthonormalbasiscanbereachedfromanyotherorthonormalbasisbysomeelementinthegroup.ThesubgroupsO(m)form

6850CHAPTER3.VECTORSPACESANDTENSORPRODUCTSUnlessrequiredbythespecificityofatopic,weshalloftenuseterminologythatdoesnotmakeexplicitwhethertheorthogonalmatricesweshallbedealingwitharespecial(determinant+1),orgeneral(also-1).nnNoticethatthemapV×V→Risnotnecessarilypreservedbytheisomorphismofallvectorspacesofthesamenumberofdimensions.Forinstance,consideranEuclideanandapseudo-Euclideanvectorspaces,bothofdimension2.Wecanchooseonebasis,respectivelyorthonormalandpseudo-orthonormal,ineachofthesetwospaces.Weestablishaone-to-onecorrespondencebetweenthevectorsofbothspacesbyidentifyingeachvectorinonespacewiththevectorthathasthesamecomponentsintheotherspace.Thescalarproductofthetwovectorsisnotingeneralthesamenumberasthescalarproductoftheirhomologuevectorsintheotherspace.Inparticular,homologuevectorsneednothavethesamenorm.3.4.3ReciprocalbasesForsimplicity,letusconsiderproperlyEuclideanspaces.Givenabasis(ai),consideranotherbasis(bj)suchthatai·bj=δij,∀(i,j).(4.9)Wesaythatthetwobasesarethereciprocalofeachother.Wedenotethisrelationbyrepresenting(b)as(aj).Thus:jjjai·a=δi.(4.10)Eventually,asweconsiderspecificstructuresthepositionoftheindiceswillacquiremeaning.The(aj)isthencalledthereciprocalbasisofthe“original”basis(ai).Givenabasis(a),Eqs.(4.10)canbeusedtoobtain(aj)intermsof(a).iiItissuggestedthatjuniorreadersexplicitlysolveEqs.(4.10)forajwhenthebasis(ai)isgivenasi+ja1=i,a2=√.(4.11)2Noticethata1anda2arenormal(magnitudeone)butnotorthogonal.WenowshowthatEqs.(4.10)uniquelydetermine(ai)intermsof(a).Ifi(ai)exists,wecanexpanditintermsoftheoriginalbasis:ai=Dija.(4.12)kIfwesubstitutethisin(4.10),weobtainjkjDgki=δi,(4.13)whichshowsthatthematrixwithentriesDjkistheinverseofthematrixg,whichalwayshasaninverse.Hence{ai}=g−1{a},(4.14)j

693.4.EUCLIDEANVECTORSPACES51wherecurlybracketdenotescolumnmatrix.Thisequationcanbeusedforobtainingthereciprocalofthebasis{ai}in(4.11)byfirstobtaininggij=ai·ajandtheninvertingthismatrix.Thisisanexpedientwayofobtainingreciprocalbasesifoneisefficientatinvertingmatrices.Wehaveusedandwillusecurlybracketstoemphasizethattheelementsofthebasesarewrittenascolumnswhenasquarematrixactsonthemfromtheleft.InparalleltothedefinitionofgijinEq.(4.1),wedefinegij≡ai·aj=gji.(4.15)Substitutionof(4.12)in(4.15)yields:gij=DilDjka·a=DilgDkj,(4.16)lklkwherewehavemadeuseofthesymmetryofDij,i.e.ofthematrixg−1.Equa-tions(4.13)and(4.16)togetherimplygij=δiDkj=Dij(4.17)kand,therefore[gij]=g−1,(4.18)where[gij]denotesthematrixofthegij.Hence,werewriteEq.(4.14)asai=gija(4.19)jwhichinturnimplies,premultiplyingbygkiandcontracting:gai=ggija=δja=a.(4.20)kikijkjkEquations(4.19)-(4.20)statethatwemayraiseandlowerindicesofbasisvectorswithgijandgrespectively.Itmustbeclearfromthedefinition(4.17)howijthegijtransformunderachangeofbasis.Inthesamewayaswehaveintroducedcomponentsviofavectorwithrespecttothebasis(ai),wenowintroduceitscomponentsviwithrespecttothebasis(ai).Thusv=via=vai.(4.21)iiThebasiswithsubscriptswillbecalledcontravariantandsowilltheaccompa-nyingcomponents,i.e.thevi’s.Thebases(ai)andthecomponents(v)willbeicalledcovariant.Thedotproductofvandajyields,using(4.21),iiv·aj=(via)·aj=viδj=vj.(4.22)Hencev=v·a=(via)·a=vig.(4.23)jjijijOneequallyshowsthatweraiseindiceswithgij.Hencetherulesforraisingandloweringtheindicesofthecomponentsarethesamerulesasforthebasiselementswithindicesofthesametype.

7052CHAPTER3.VECTORSPACESANDTENSORPRODUCTSLetAdenotethematrixthattransforms(a)into(a).Onereadilyshowsiithatthematricesgandgobtainedfromtherespectivebases(a)and(a)areiirelatedbyg=AgAT.(4.24)Weinvert(4.24)andobtain−1T−1−1−1T−1−1T−1Tg=(A)gA=(A)g[(A)].(4.25)thesymmetryrelationexistingbetweenabasisanditsdualallowsustoimme-diatelyinferfromEqs.(24)-(25):iT−1i{a}=(A){a},(4.26)aresultwhichisobviousinanycaseinviewofpreviousconsiderations.Exercise.Considera(pseudo)-Euclideantwodimensionalvectorspacesuchthatg=1,g=−1andg=g=0incertainbasis{ai},i=1,2.Find11221221ginanotherbasis{a}suchthatanarbitraryvectorv=aa+babecomes:iji12v=a(1−ρ2)1/2a+(b−ρa)(1−ρ)1/2a,(4.27)12whereρisaparameterenteringthechangeofbasis.(Yourresultmustcontainonlynumbersandtheparameterρ.)Thetransformationofthegij’sunderachangeofbasesisobvious:g=a·a=AklklijijiAjak·al=AiAjgkl,(4.28)andsimilarly,butnotidentically,forthetransformationofthegij.Wereturntotheopeningstatementofthissection,witharemarktokeepgoinguntilweaddressEuclideanspacesinchapter6.ThesurfaceofatableextendedtoinfinityandthespaceofordinarylifeasconceivedbeforetheadventofthetheoriesofrelativityaresaidtobeEuclideanspacesofdimensions2and3respectively.TheyarenotEuclideanvectorspaces,orevensimplyvectorspacesforthatmattersincetheydonothaveaspecialpoint,azero.Ifwechoosesomearbitrarypointtoplaytheroleofthezero,wemaythenuseinEuclideanspaces,thetoolsofEuclideanvectorspaces.3.4.4OrthogonalizationTworelatedprocessesoforthogonalizationTherearetworelatedmainproblemsoforthogonalization.Oneofthemis:givenavectorbasisthatisnotorthonormal,findintermsofitanotheronethatis.AdirectwaytosolveitisthroughthesocalledSchmidt’smethod.Thesecondproblemis:givenaquadraticform,reduceittoasumofsquares.WeshallsolveitthroughadirectmethodthatwefoundinCartan[23].Itcould

713.4.EUCLIDEANVECTORSPACES53alsobesolvedindirectlythroughitsrelationtothefirstproblemsincethetwoproblemsareinessencethesame,asweshowinthethirdsubsection.Theissueisessentiallyaboutorthogonality,sincenormalizingistrivialoncewehaveorthogonality.Forthisreason,wehavesofarspokenonlyoforthogo-nalizationinthetitlesofthesectionandofthissubsection.Schmidt’sorthogonalizationprocedureGivenanybasisofaspaceEn,onecanalwaysfindnindependentlinearcom-binationsofitselementssuchthattheselinearcombinationsconstituteanor-thonormalbasis.Ofcourse,thesolutionisnotunique,since,oncewehavegeneratedoneorthonormalbasis,wecangeneratealltheothersbyactionoforthogonalmatrices.Thebasicproblemisthusthegeneratingofjustoneor-thonormalbasis,whichSchmidt’smethodachieves.Inordertominimizeclutterinthecomputationwithsymbolsthatfollows,itisbesttofirstorthogonalizethebasisandonlythennormalizeit(asopposedtonormalizingaftereachstepofdiagonalization).Weassumethedotproductseμ·eμtobeknown,includingwhenμequalsν(ifnotso,whatwoulditmeantoknowthebasis?).Webuildanorthogonalbasis(aμ)asfollows:a1≡e1,(4.29)1a2≡e2+λ2a1,(4.30)12a3≡e3+λ3a1+λ3a2,(4.31)123a4≡e4+λ4a1+λ4a2+λ4a3,(4.32)andsoon.Noticethat,exceptforthefirsttermontherighthandoftheseequations,alltheothersinvolvetheaμ’s,nottheeμ’s.Inthisway,wemakemaximumuseoforthogonality.Theλ’swillbedeterminedsothat(aμ)beorthogonal.Dot-multiplying(4.30)bya1,weget10=e2·a1+λ2,(4.33)1whichallowsustoobtainλ2.Nextwedot-multiply(4.31)bya1.Wealsomultiply(4.31)bya2.Wethusget10≡e3·a1+λ3,(4.34)20≡e3·a2+λ3,(4.35)12whichdetermineλ3andλ3.Noticethatweknewatthispointwhata1anda2areintermsofe1ande2,sothatwecancomputethedotproducts.We1nowmultiply(4.32)bya1,a2anda3toobtainthreeequationsthatyieldλ4,23λ4andλ4.Andsoon.Finally,asstated,weorthonormalizetheelementsaμproceedinginthesameorderinwhichtheywereobtained.

7254CHAPTER3.VECTORSPACESANDTENSORPRODUCTSOrthonormalizationofbasesandreductionofquadraticformsLetusreferavectorvtothebases(eμ)and(ˆaμ),whereˆaμresultsfromthenormalizationofaμ:v=vμe=Vνˆa.(4.36)μνWecouldalsohaveusedontherighthandsidetheindexμ,butinexperiencedreadersmayinadvertentlythinkthatthisisanequalitytermbyterm.Wethenhave:v2=vμvνg=(V1)2+(V2)2+...+(Vn)2.(4.37)μνHencetheproblemoforthonormalizingabasisisequivalenttotheproblemofreducingasymmetricquadraticformtotheCartesianform(Vμ)2.(4.38)μThisreductionisequivalenttofindinglinearcombinationsVμ’softhevν’ssothat,whenwesubstitutethemin(4.38),werecovervμvνg.μνRemarkconcerningbasesofspacesthatarenotproperlyEuclideanAssumea2-DLorentzianspace.Considerabasis(a,a)suchthatˆa2=1,010ˆa2=−1andˆa·ˆa=0.Defineanotherbasis,(e,e),as101+−ˆa0+ˆa1ˆa0−ˆa1e+≡√,e−≡√.(4.39)22Wethenhavee2=0,e2=0,e·e=1.(4.40)+−+−Theequationin+−P≡tˆa0+xˆa1=ye++ye−,(4.41)allowsustorelate(t,x)to(y+,y−)byusing(4.39)in(4.41),+ˆa0+ˆa1−ˆa0−ˆa1tˆa0+xˆa1=y√+y√,(4.42)22thusobtainingy++y−y+−y−t=√,x=√.(4.43)22Hence,22+−P·P=t−x=2yy.(4.44)InotherwordsP·P=gyμyν,(4.45)μνwithy1≡y+,y2≡y−(4.46)andg11=g22=0,g12=g21=1.(4.47)Wehavethusseenthat,whenaquadraticexpressionisnotpositivedefinite—(t2−x2)isnot—,wecanhavebaseswherealltheg(diagonalterms)arezero.μμ

733.5.NOTQUITERIGHTCONCEPTOFVECTORFIELD55Cartan’sreductionofaquadraticsymmetricformtoasumofsquaresWehaveseenthatdiagonalizationandorthonormalizationaretwoaspectsofthesameproblem.Weshallnowbeinterestedindirectlydiagonalizingagivensymmetricquadraticform.Theprocessweareabouttoconsiderapplieswhenatleastoneofthegμμisnotzero.Thatwillbetheonlycasewearehereinterestedin,sincethereisnotsignificantmotivationthatweknowoftoconsiderthealternativecase.Readersinterestedinthecaseg11=g22=...=gnn=0shouldrefertoCartan[23].Letg11denoteagμμdifferentfromzero.Leta,b=(2,3,...n),...Thequadraticformabμν112n2gabxx≡gμνxx−(g11x+g12x+...+g1nx)(4.48)g11doesnotcontainthevariablex1.Define1μy≡g1μx,(4.49)andrewrite(4.48)asμν112abgμνxx=(y)+gabxx.(4.50)g11Wewouldnowproceedinthesamewaywithgxaxb.Endofproof.abThemetriconasurfacedependsononlytwoindependentcoordinates,usu-allycalledparameterstodistinguishthemfromthecoordinatesofthe3-DEu-clideanspaceinwhichweconsiderthesurfacetobeembedded.Letxλrefertothesetwocoordinates.Equation(4.50)thenreducestoasumofsquares:12μνy22gμνxx=√+g22x.(4.51)g11ThiswillbeusedinappendixA.3.5NotquiterightconceptofVECTORFIELDManyreaderswillhavecomeacrosstheconceptoftensorfieldasasetoffunc-tionsthatunderachangeofcoordinatestransforminsuchasuchaway.Thetransformationinvolvespartialderivativesofonesystemofcoordinateswithrespecttoanother.Thisisahorribleapproachtotheconceptoftensorfields,andinparticulartovectorfields(whicharetensorfieldsof“rankorgradeordegreeone”).Onecancertainlyactinreverseifcircumstancessoadvice,byendowingsomesetofquantitieswithrulesthatmakeitbecomeavectorortensorspace.Onemustthenunderstandthestructurethat,insodoing,onewouldbecreating.Toavoidclutterandmakethingssimple,weonlydiscussvectorfields.Noticehowwedefinedvectorsinsection2.Wefirstdefinedasetwithappropriateproperties.Vectorsaresimplythemembersofthosesets,now

7456CHAPTER3.VECTORSPACESANDTENSORPRODUCTSstructures.Thedefinitionofthepreviousparagraphdoesnotmakereferencetothestructuretowhichtensorandtensorfieldsbelong.Ithastobeinferredfromthespecificapplicationthatonemakesofthosetensorfields,ifpossible.Oneshouldfirstdefinethetensorproductofvectorspaces.Tensorsarethemembersofthoseproductspaces,likevectorsarethemembersofvectorspaces.Inthissection,weareinterestedinvectorfields.Avectorfieldonsomemanifold,region,surface,etc.isavectorateachpointofthemanifold,region,surfaceetc.Thevectorsatdifferentpointswillingeneralbelongtodifferentvectorspaces.Thatiswheredifferentialgeometrybecomesinteresting.Whenspecializedtocontravariantvectorfields,thedefinitionoftheopeningstatementofthissectionreads:“acontravariantvectorfieldisasetofquantitiesvithatunderachangeofcoordinatestransformslike∂xjvj=vi.”(5.1)∂xiThisexpressionrepresentshowthecomponentsofatangentvectorfieldinonevectorbasisfieldarerelatedtothecomponentsofthesamevectorinanothervectorbasisfield.Ateachpoint,the∂xj/∂xiarevaluesofcoordinatesAiinjstructuresthatweshallintroducelaterunderthenamesofframebundlesandbundlesofbases.Inthoseveryimportantstructures,theAiareadditionaltojandindependentofthexcoordinates.Thevectorfieldscomponentsin(5.1)arenotsorelativetogeneralbasisfields,butto“coordinatebasisfields”oftangentvectors.Theconceptoftangentvectorwillbeintroducedinduetime.Fromthedefinition(5.1),itisnotclearwhatthefieldisafieldof.Theprecedingconsiderationswillbeunintelligibletomanyreaders.Wehavenotyetdevelopedthemachinerytounderstandwithexampleswhy(5.1)isawrongconceptofcontravariantvectorfield,sincetherearealsootherentitieswithcomponentsthattransforminthesameway.Itismuchsimpler—becausewehaveclearerexamplesatthispoint—todiscussthedefinitionofcompo-nentsofcovariantvectorfieldsgiveninthesamepublicationswhere(5.1)isintroduced.Theyaresetsofquantitiesλithatunderachangeofcoordinatestransformlike∂xjλi=∂xiλj.(5.2)Thiscorrectlystateshowthecomponentsofacotangentvectorfield(alsocalledcovariantvectorfield)transformunderachangeofcoordinatefieldofbases.But(5.2)alsoapplytothecomponentsofourdifferential1−forms,whicharefunctionsofcurves,notfunctionsofvectors.Inaddition,ifthevectorspacenVisEuclidean,wemightbereferringtoatangentvectorfield(alsocalledcontravariantvectorfield)intermsofreciprocalbases.Hence,howsomethingtransformsisnotanindicatorofthenatureofanobject,unless,ofcourse,oneisdealingwiththebluntinstrumentofthetensorcalculus,wherethereisapoorerpaletteofconcepts.Noticethatwecriticizedtensorcalculus,nottensoralgebra.Theformeriscontranatura.Thesecondispresentalmostanywherein

753.6.TENSORPRODUCTS:THEORETICALMINIMUM57differentialgeometry,indirectlythroughquotientalgebras(seealittlebitmoreonthisinthenextsection)..Tosummarize:(5.1)doesnotreferonlytocontravariantvectorfields.Whenitdoes,itcharacterizesonlysetsofcomponentsofvectorfieldsrelativetoarbi-trarybasesofthese.3.6Tensorproducts:theoreticalminimumInthisbook,weshallrestrictourselvestoaminimumofconceptsrelatedtoten-sors,thenextsectionbeinganexception.Asweareabouttosee,readershavealreadybeendealingwithtensorproductswhenmultiplyingdifferentstructuresratherthandifferentcopiesofonestructure.Theseproductsaresonaturalthattheygounnoticed.Thatfamiliarityisalmostsufficientforourpurposes.Considertheexpressiondxi+dyj+dzk,alsowrittenasjiδidxaj.(6.1)Itisamemberofthetensorproductofamoduleofdifferential1−formswitha(tangent)vectorspace.Themoduleisasubmoduleofthealgebraofdifferentialforms,itselfamoduleasanyalgebraisinthefirstplace.The(aj)retrospectivelyrepresent,asweshallsee,afieldofvectorbaseswhich,intheparticularcaseofthe(i,j,k),simplyhappenstobeaconstantbasisfield.Inasmuchasweknowhowtohandlethesequantities,weknowhowtodealwithtensorproductsofdifferentstructures.Themorecommonversionofthetensorcalculusdoesnotmakeexplicituseofbases,onlyoftheircomponents;jthus(6.1)wouldbegivenasδi,inthattensorcalculus.Onedoesnotneedtoknowhowcomponentstransformifthenewcomponents(intermsoftheoldones)canbereadfromtheexpressionthatresultsfromsubstitutionoftheoldbasisintermsofthenewoneinexpressionssuchas(6.1).Noticealsothatwecanchangejustthebasisofdifferentialforms,orjustthebasisoftangentvectors.Thisisnotthecaseinthemostcommonversionofthetensorcalculus,wherebothchangesgotogether.Thefreedomjustmentionedisanadvantageofmakingthebasesexplicit.Sincetheexpressionδjdxiaisavector-valueddifferentialform,itisclearijwhyanin-depthunderstandingofwhatvector-valued(and,ingeneral,tensor-valued)differentialformsareinvolvestensorproducts.Butalackofmajorimmersionintensoralgebrainthisbookwillnotproducealargelossinunder-standingofthetopicsofpracticalinterest.Forthisreason,thenextsectioncanbeignoredbyreaderstowhichtheseconceptsaretotallyforeign,unlesstheyaretrulycuriousand/orchallenged.Anotherwayinwhichtensoralgebraisinvolvedinthisbookisthatexterioralgebra,asinthepreviouschapter,isasocalledquotientalgebraofthegen-eraltensoralgebraconstructeduponthemoduleofdifferential1−forms.Thegeneraltensoralgebraforagivenvectorspaceisthealgebraofalltensorsofallranks.Quotientalgebraisapowerfulconceptbuttoosubtleforthosenotveryknowledgeableinalgebra.Itneednotbedealtwith,providedweknow

7658CHAPTER3.VECTORSPACESANDTENSORPRODUCTShowtoworkinit.Wemustunderstand,however,thatquotientalgebrasarenotsubalgebras,sincetheproductinsubalgebrasremainthesameasintherespectivealgebras.Inthequotientalgebras(exterior,Clifford),ontheotherhand,theproductsaresomethingdifferent,evenifitsobjectslooklikethoseinthemotheralgebra.Inordertoconnectwiththecontentsofthepreviousparagraph,letusstatethattensor-valueddifferentialformsaremembersofastructurewhichisthetensorproductofsometensorspacebythealgebraofscalar-valueddifferentialforms.Butthisstatementneedstoberevisitedaswegodeeperanddeeperintodifferentialgeometry.Inthetensorcalculus,theconceptofconnection,whichiscentralinYang-Millsdifferentialgeometry,isundervalued.Aperniciouseffectofthisisthelackofagoodconceptofcurvatureinmoderndifferentialgeometryforphysicists.BothconnectionandcurvatureareaslegitimateLiealgebravalueddifferentialformsin“non-Yang-Mills”asinYang-Millsdifferentialgeometry.Withthisperspective,SU(2)Yang-MillstheoryisidenticaltotheclassicalEuclideange-ometryofalgebraicspinorsinthreedimensions(seechapter6).TheissueiswhethertheLiealgebrasu(2)thatappearsinnon-Yang-Millsdifferentialgeom-etryhasanythingtodowiththesu(2)algebraofYang-Millstheory.Onehasimplicitlyassumedintheliteraturethattheanswerisinthenegative,butthismightonedaybeviewedasanincorrectinferencedoneatthetimewhentheLiealgebrasemerginginnon-Yang-Millscontextarelargelybeingoverlookedorignoredbecauseofthetensorcalculus.3.7FormalapproachtoTENSORS3.7.1DefinitionoftensorspaceThissectioncanbeignoredwithoutmajorconsequence.Ontheotherhand,readerswhoareinterestedinknowingevenmoreaboutthissubjectmayrefertoabeautifulbookbyLichnerowicz[53].mnm×nmConsidertheabstractvectorspacesU,V,andW.Letu∈U,nm×nv∈Vandw∈W.Consideramap(u,v)→w,whichweshalldesignateas(u,v)→u⊗v,withthefollowingproperties:(a)u⊗(v1+v2)=u⊗v1+u⊗v2,∀(u,v1,v2),(u1+u2)⊗v=u1⊗v+u2⊗v,∀(u1,u2,v),(b)ifαisanarbitraryscalar:(αu)⊗v=u⊗(αv)=α(u⊗v),(c)if(a1,...,am)and(b1,...,bn)designatearbitrarybases,respectivelyofmnUandV,thentheai⊗bα,(i=1,...,m,α=1,...,n)constituteam×nbasisciαofthevectorspaceW.m×nWethensaythatthevectorspaceWis(canbeviewed,becomes)themnmntensorproductU⊗VofthevectorspacesUandV.Thevectorsof

773.7.FORMALAPPROACHTOTENSORS59m×nmnW,whenconsideredaselementsofU⊗V,arecalledtensors.Pedes-trianapproachestothetensorcalculusbasedonbehaviorundercoordinatetransformationsjustmissthealgebraicnatureoftheseobjects.Alinearcombinationoftensorproductsisatensor,buttheconverseisnottrue:sumsoftensorsmayormaynotbewrittenasatensorproducts.Asumoftensorsisatensorandmultiplicationofatensorbyascalarisanothertensor.Properties(a)and(b)arefamiliarproperties.Lessfamiliarandmoresubtleis(c).Howcoulditnotbesatisfied?Thedot,exteriorandvectorproductsdomnnotsatisfy(c).SupposeforsimplicitythatUandVwerejusttwocopiesof22atwodimensionalvectorspace,V,andlet(ai)beabasisinV.Wehavea1∧a1=a2∧a2=0,a1∧a2=−a2∧a1.(7.1)Noticethatthesubspaceofthoseproductsforthisparticularexampleisnow1-dimensional,ratherthan4-dimensional.Thisproduct,theexteriorproduct(whichweusedfordifferentialforms),isnotatensorproduct.Foranotherexample,wecouldhavea1∨a1=a2∨a2=1,a1∨a2=−a2∨a1.(7.2)Thealgebraof∨isonewheretheproductoftwovectorsisconstitutedbythesumoftheirexterioranddotproducts.Inthe∨−algebra,therearesubspacesofscalars,vectors,bivectors,stoppingatmultivectorsofgraden.The“1”in(7.2)istheunitscalar.Weshallnotentertodiscussthisproductfurther,asitinvolvesnewsubtleties.Fromtheaxiomsonereadilyobtainsthatiαu⊗v=uvciα,(7.3)whereu=uiaandv=vαb.Viceversa,onecanshowthatthiscompositioniαlawistheonlyonethatsatisfiesallthreeaxioms.3.7.2TransformationofcomponentsoftensorsIndifferentialgeometry,weshallencountertensorproductsu⊗vwhereuandvbelongtotwodifferentvectorspaces.Atthispoint,letusconcentrateonnntensorswhichresultfromtensorproductV⊗Vofavectorspacebyitself.nnWeshallusethetermsecondranktensorstorefertotheelementsofV⊗V.Asecondranktensor,alternativelycalled“ofrank2”,canbewrittenasijT=Tai⊗aj.(7.4)nIfwechooseanewbasisinV,thesametensorcanbewrittenasijijT=Tai⊗aj=Tai⊗aj,(7.5)withTij≡TijandainordertofacilitateoperationswheretheAiori≡aiktheAikareinvolved.Substitutionof(2.3b)in(7.4)andcomparisonwith(7.5)yieldsTkl=AkAlTij,(7.6a)ij

7860CHAPTER3.VECTORSPACESANDTENSORPRODUCTSwherewehaveremovedonthelefttheredundantuseofprimedindices.Simi-larly,substitutionof(2.3a)in(7.5)andcomparisonwith(7.5)yields:Tij=Aijkl(7.6b)kAlTInobtainingEqs.(7.6),wehaveusedthatthe(ai⊗aj,∀i,j≤n)andthe(a⊗a,∀i,j≤n)constitutebasesofVn⊗Vn.ijNoticethat,inEq.(7.6a),n2quantitiesAisufficetodescribethetransfor-knnmationofthetensorsbelongingtoV⊗V.WhereasthelineartransformationsofVn×naregivenbyalltheinvertiblen2×n2matriceswhichconstitutethegroupGL(n2)ofn4parameters,inVn⊗Vnoneonlyconsidersasubgroupofallthesepossibletransformations.Itonlyhasn2parametersAi=1...n),k(i,kthetransformationsnowbeingquadraticintheseparameters.Asecondranktensorfieldisatensorateachpointinsomedomain.Inthiscase,wehaveTij(x)=Aij(x)Tkl(x)(7.7)k(x)Alwherexdenotesanynumberofcoordinatesinagivenspace.WeshallseeinkfuturechaptersthattheAii/∂x.Whenthatisthek(x)couldtaketheform∂xcase,thelastequationbecomes∂xi∂xjTij(x)=Tkl(x).(7.8)∂xk∂xlThisformulaisusedforthetypeofdefinitionoftensorfieldtowhichwereferredinsection5.Itappliesonlytocertaintypesofbases,theonlybasesthatareusedinpedestrianapproachestothetensorcalculus.Bysuccessiveapplicationoftheconceptoftensorproductoftwospaces,mnonecangeneratethetensorproductofseveralvectorspaces.SinceU⊗Vmnpmisavectorspaceitself,wedefineU⊗V⊗Xasthetensorproduct(U⊗npV)⊗X.Thiswillinturnbeavectorspaceofdimensionm×n×p.WemnpcouldalsohavedefinedthetensorproductU⊗(V⊗X),whichisalsoavectorspaceofdimensionm⊗n⊗p.Let(ai),(bα)constituterespectivebasesmnpofU,VandX.Weshallidentify(ai⊗bα)⊗cAwithai⊗(bα⊗cA),andshallrefertotheseproductsassimplyai⊗bα⊗cA.Inthesameway,wedonotmnpneedtheparentheseswhenreferringtothetensorproductofU,VandX.mnpm×n×pHencethetensorproductU⊗V⊗XisthevectorspaceVinwhichweonlyconsiderthosechangesofbasesintroducedbythechangesofbasesinmnpU,VandXthroughthemapping(u,v,x)→(u⊗v⊗x).Tensorsofrankrthataremembersofthetensorproductofavectorspacebyitself(rfactors)canbewrittenasi1...irT=Tai1⊗...⊗air.(7.9)nnIfwechooseanewbasisforV,weinduceachoiceofanewbasisforV⊗nV⊗...,andthesametensorcannowbeexpressedasT=Ti1...ira⊗...⊗a.(7.10)i1ir

793.8.CLIFFORDALGEBRA61Substitutionof(2.3a)in(7.10)andcomparisonwith(7.9)yieldsTi1...ir=Ai1...AirTj1j2....jr.(7.11)jj1rSimilarly,substitutionof(2.3b)in(7.9)andcomparisonwith(7.10)yieldsTi1...ir=Ai1...AirTj1j2...jr.(7.12)j1jrnnNoticeagainthatthetransformationsinVyieldthetransformationsinV⊗nn2iV⊗...⊗V.ThenindependentquantitiesAjdescribethetransformationsoftensorofanyrank.Oncemore,scalars(whichusuallyarerealorcomplexnumbers)andvectorsaresaidtobeofrankszeroandonerespectively.Wespeakofrank.Itwouldbemoretechnicaltospeakofgrade,sincethesetofalltensorsofallgradesandtheirsums(generallyinhomogeneous,meaningthatthegradesaremixed)isasocalledgradedalgebra.Wedoaswedoinordertoemphasizethatdifferentialformsarenottensors.Usingthetermsrankfortensoralgebraandgradeforquotientalgebraswillhelpustorememberthat.n∗nInalloftheabove,VcouldhavebeenreplacedwithV.Andbothinturncouldbereplacedwithfieldsofsuchvectorspaces,oneofeachateachpointofsomemanifold.Andthesefields(samenotationasthevectorspacesthemselves)canbetensormultipliedbyamoduleofdifferentialforms.Thus,onewayof∗nnlookingatcurvaturesisaselementsofthetensorproductofV⊗Vbyamoduleofdifferential2−forms(Thereisalsoaconceptofcurvaturethatisan∗nn∗n∗nelementofV⊗V⊗V⊗V).WeshalllaterseewhattheconceptofLiealgebravaluednesshastodowiththis.Algebrasandidealsalsocouldbefactorsintensorproductofstructures,whichiseverywhere.Andwedonotneedtobeawareofthembecausetheirdefiningproperties(a)and(b)arealltoonatural.3.8Cliffordalgebra3.8.1IntroductionSomereadersmightconsiderreadingthissectionjustbeforesection6.4.Bythen,theywillhaveacquiredpracticewithexterioralgebra,whichbringsthemclosertoCliffordalgebra.Weheregiveaslightideaofanassociativealgebra,calledCliffordalgebra,thatgoesbeyondtheexterioralgebraonwhichtheexteriorcalculusisbased.Itwillbeusedinsection6.4.Whenitmakesoccasionalappearanceindifferentialgeometry,itspresenceisdisguised,takingtheformofadhocconcepts.Inthetensorcalculus,theLevi-Civitatensorisonesuchconcept.Onecontractsitwithothertensorsinordertoobtainwhat,inCliffordalgebra,arecalledtheirHodgeduals.OneusefulpurposeofHodgedualityisthatitpermitsonetoobtaininteriorderivatives(followingK¨ahler,thetermderivativeisnotrestrictedonlytooperationssatisfyingthestandardLeibnizrule)bycombiningHodge

8062CHAPTER3.VECTORSPACESANDTENSORPRODUCTSdualitywithexteriordifferentiationinordertoobtain“interiordifferentiation”.ThisconceptualreplacementisonlypossibleinEuclidean,pseudo–Euclidean,Riemannianandpseudo-Riemannianspaces.Thevectorproductand,therefore,vectoralgebraarepeculiaritiesof3-DEuclideanvectorspace(intherestofthisbook,EuclideanandRiemannianwillalsomeanpseudo-Euclideanandpseudo-Riemannian,exceptwhenindicatedotherwise).Theydonotexistinarbitrarydimension.Allthosespaces,however,haveaCliffordalgebraassociatedwiththem,differentfromonesignaturetoanother.Avectorproductisthecombinationofaexteriorproductandthedualityoperation.Exceptinthemostgeneralcases(meaningwhenbothfactorsinaproductareofgradetwoorgreater),Cliffordproductisthecombinationoftheexteriorandinteriorproduct(readdotproductifyouprefer).Thevectorproductdoesnotcombinewiththedotproduct.Thatisoneofitshandicaps.Manyelementsofthealgebrahaveinverses.Amongthosethatdonot,mostinterestingaretheelementsthatareequaltotheirsquares(and,therefore,toalltheirintegerpowers).Theyarecalledidempotents.Otherimportantconceptsarespinors,whicharemembersofidealsinthealgebra.Inviewofwhathasbeensaidinthelasttwoparagraphs,weshouldconsiderCliffordalgebraasthetrue,natural,canonical,proprietary,definingalgebraofspacesendowedwithametric.Examplesfollow.ComplexalgebraistheCliffordalgebraofone-dimensionalpseudo-Euclideanvectorspace.QuaternionsconstituteanalgebraisomorphictotheCliffordalgebraof2-DEuclideanspacewithsignature(-1,-1),althoughitisformulatedasifitwereassociatedwiththreedimensions.ThealgebrasofthePauliandDiracmatricesarethealgebraof3-DEuclideanand4-DLorentzianvectorspacesrespectively.CorrespondingtoaEuclideanvectorspaceofdimensionn,thereisaCliffordalgebrawhich,asavectorspace,isofdimension2n.Thus,thePaulialgebraisaspaceofdimension8.AbasisofthealgebraisconstitutedbythematricesI,σ1,σ2,σ3,σ1σ2,σ2σ3,σ3σ1,σ1σ2σ3.(8.1)Ithappensthatthesquareofσ1σ2σ3istheunitmatrixtimestheunitimag-inary.Butitisbettertothinkoftheunitimaginarytimestheunitmatrixastheunitelementofgradethreeinthealgebra,likeσ1,σ2andσ3areunitsofgradeoneandσ1σ2,σ2σ3andσ3σ1areunitsofgradetwo.Warning,wemayspeakofσ1σ2,σ2σ3,σ3σ1asunitsofgradetwoonlybecausetheyareorthogonaland,therefore,theyareequaltoσ1∧σ2,σ2∧σ3,σ3∧σ1.Exteriorproductsareofgradetwo,whetherthefactorsσiareorthogonalornot.3.8.2BasicCliffordalgebraAssumetheexistenceofaproductwhichisassociativeanddistributivewithrespecttoaddition,butnotnecessarilycommutativeoranticommutative.Weformthefollowingidentityab≡(1/2)(ab+ba)+(1/2)(ab−ba).(8.2)

813.8.CLIFFORDALGEBRA63Weintroducesymbolstonamethesetwopartsoftheproductasindividualproductsthemselves:a·b≡(1/2)(ab+ba),a∧b≡(1/2)(ab−ba).(8.3)Itisclearthata∧a=0anda∧b=−b∧a.Inanticipationofwhatweareabouttosay,wehaveintroducedthesymbolfordotproducttorepresentthesymmetricpartoftheCliffordproduct.WehavenotyetcharacterizedCliffordalgebra.Thismustbeclearwhenwerealizethattensorproductssatisfythepropertiesofbeingassociativeanddis-tributivewithrespecttoaddition,since(1/2)(ab+ba)wouldbeatwotensorifaandbwerevectors.Weassociateascalarwithit,specificallywhatweknowasthedotproductofthetwovectors.abthenistheirCliffordproduct.Ifthefore-goingsoundstooabstract,thinkoftheproductsofthe(fourgamma)matricesofrelativisticquantummechanics.TheyareCliffordproducts.(1/2)(ab+ba)thenisascalarmultipleoftheunitmatrix.Suchmultiplesplaytherolesofscalarsinmatrixalgebras.Itisobviousfromequations(8.2)and(8.3)thatab=a·b+a∧b,(8.4)whichbringstheexterioranddotproductsclosertoeachotherthanthevectoranddotproductsare.WeareinterestedinunderstandinghowbasicelementsofEuclideanvectorcalculus(wesaidEuclideaninordertohavethedotproduct)arepresentintheexteriorcalculus.Consideranytwonon-colinearvectorsina3-dimensionalsubspaceofann-dimensionalvectorspace.Weshalldenoteas(i,j,k)anor-thonormalbasisofthissubspace.Wethenhavea∧b=(a1i+a2j+a3k)∧(b1i+b2j+b3k)=(8.5)=(a1b2−a2b1)i∧j+(a2b3−a3b2)j∧k+(a3b1−a1b3)k∧i,wherewehaveusedantisymmetry.Intermsofi∧j,j∧kandk∧i,thecomponentsoftheexpressionontherighthandsideof(8.5)arethesameasforthevectorproduct,butonenolongerassociatesi∧j,j∧kandk∧iwithk,iandj.Weshallsaythata∧brepresentsoriented2-dimensionalfiguresintheplanedeterminedbyaandb,andofsizeabsinθ,whereθistheanglebetweenaandb.λ(a∧b)andλ(b∧a)representthesamefiguresbutwithoppositeorientationandmagnitude,λabsinθ.Thecontentsoftheparenthesesontherighthandsideof(8.5)aresaidtorepresentthecomponentsofthe“bivector”a∧bintermsofthebivectorsi∧j,j∧kandk∧i.Theexteriorproductitselfrepresentstwodimensionalfiguresofthesamesizeastheparallelogrambuiltwiththetwovectorsassidesandinthesameplane.Letusdenoteasaithevectorsofabasisinarbitrarydimension.Fromthefirstofequations(8.3),wehaveaiaj+ajai−2gij=0,(8.6)

8264CHAPTER3.VECTORSPACESANDTENSORPRODUCTSwheregijisthedotproductofaiandaj.Equation(8.6)istakenastheequationthatdefinestheCliffordalgebra,ofwhichmanydefinitionsexist.TheCliffordproductofavectorandamultivector(bivector,trivector,etc.)alsosatisfyarelationlike(8.4),namelyaA=a·A+a∧A.(8.7)IfAisofhomogeneousgrader,aAwillhaveingeneralapartofgrader−1andanotherpartofgrader+1.ACliffordalgebramaybeconstructednotonlyuponavectorspacebutalsouponamoduleofdifferential1-forms.Inthatcase,thealgebrareceivesthespecificnameofK¨ahleralgebra.Ishallsometimesusethesymbol∨forCliffordproduct,insteadofjuxtaposition;otherwisedxdy,meaningdx∨dy,couldbeconfusedwithdxdyundertheintegralsign.Inparallelwith(8.6),thedefiningrelationoftheK¨ahleralgebraof“clifforms”isdxμ∨dxν+dxν∨dxμ=2gμν.(8.8)3.8.3ThetangentCliffordalgebraof3-DEuclideanvectorspaceAlittlebitofpracticewithin3-DEuclideanspacewithouttheuseofmatriceswillbehelpfultogetthegistofthisalgebra.Inordertoemphasizethatmatricesdonothavetodowithitsessence,wedonotusethesymbolsσibutrathernotationtypicalofgeometry.Becauseoforthogonality,wehavei·j=0,andi∨j=i∧j.(8.9)Ontheotherhand,i∨i=i·i.(8.10)MembersoftheCliffordalgebraof3-DEuclideanthreespacecanbeex-pandedintermsofthefollowingbasisofunitelements:1,i,j,k,j∧k,k∧i,i∧j,i∧j∧k,(8.11)whichisnothingbut(8.1).Agenericnamefortheelementsin(8.11)ismulti-vectors.MoreimportantthanevenliterallyreproducingstandardvectoridentitiesinthislanguageistheknowledgeofwhatexpressionsplaythesameroleintheCliffordalgebra.Thus,forinstance,thevolumescalaru·(v×w)isreplacedwiththetrivectoru∧v∧w,(8.12)whichisnotascalar.Tobeavolumeistobeatrivector,liketobeanareaistobeabivector.

833.8.CLIFFORDALGEBRA65Asanotherexampleconsiderthevectoridentityu×(v×w)=v(u·w)−w(u·v).(8.13)WhatmattersisnotitstranslationtoCliffordalgebra,butratherthatitsroleisplayedbyu·(v∧w)=(u·v)w−(u·w)v,(8.14)whichisaparticularcaseoftheruletodot-multiplyavectorbyamultivectoru·(v∧w∧r...)=(u·v)w∧r..−(u·w)v∧r...+(u·r)v∧w...−...,(8.15)withalternatingsigns.AnimportantcontributionofCliffordalgebraisitstreatmentofrotations,whichissimilarforvectors,generalmembersofthealgebraandspinors.Amultiplicationontheleftbyacertainexponentialrotatesaleftspinor.ThesamemultiplicationfollowedbymultiplicationbytheinverseexponentialontherightrotateselementsintheCliffordalgebra.Twotheoremsof3-DEuclideangeometry(namelythatreflectionsofavectorwithrespecttotwoperpendiculardirectionsinthesameplaneareopposite,andthattheproductoftworeflectionsisarotation)allowonetoreadilygetthegeneratorsofSU(2)asthebivectorsofthetangentCliffordalgebraof3-space.3.8.4ThetangentCliffordalgebraofspacetimeThissubsectionismeantforreadersfamiliarwithgammamatricesbutnotwithCliffordalgebra.TheCliffordalgebraofspacetimeisdefinedbyγμ∨γν+γν∨γμ=2ημν,(8.16)whereημν=(1,−1,−1,−1).WeproceedtostatewhatremainstobedonewiththesymbolsforthosealgebrasinordertoworkwiththemaswedoinCliffordalgebra.Wefurtherdefine1γμ∧γν≡(γμ∨γν−γν∨γμ),(8.17)2and1γμ·γν≡(γμ∨γν+γν∨γμ),(8,18)2wherewehaveusedthesymbol∨inordertoemphasizethatwearedealingwithunitvectorsandnotmatrices.Obviouslyγ1γ2=γ1∧γ2+γ1·γ2.(8.19)Becauseoforthonormality,weclearlyhaveγμγν=γμ∧γν=−γνγμforμ=ν=0;γμγμ=γμ·γμ=ημμ.(8.20)Theserulesfordealingwithgammamatriceswillbetooobviousforpractitioners.

8466CHAPTER3.VECTORSPACESANDTENSORPRODUCTS3.8.5ConcludingremarksTheCliffordproductsupersedestheexterioronebecauseitprovidestheconceptofinverseofavector,whichtheexteriorproductdoesnot.Letabeavector,oravectorfield.Thedotproductofabyitselfpermitsonetodefineitsinverse,a−1,namelya/a2,whereaa≡a·a=a2.Cliffordproductsofvectorsalsohaveinverses:(abc...)−1is...c−1b−1a−1.Ifwehadexteriorandotherproducts,wewouldfirstexpressthemintermsofCliffordproductsinordertothenfindwhethertheyhaveinverses.Ofgreatinterestaretheinhomogeneousmultivectorsthatareequaltotheirsquares.Calledidempotents,theygenerateideals.Membersoftheseidealsarespinors.Inadditiontothemathematicalpre-eminenceofCliffordalgebraoverexte-rioralgebra(andsimilarlyforcorrespondingcalculi),therearephysicaladvan-tages.Whereasexterioralgebraisalmostsufficientfordifferentialgeometry,quantummechanicsrequiresCliffordalgebra.

85Chapter4EXTERIORDIFFERENTIATION4.1IntroductionArigoroustreatmentofallthingspertainingtoscalar-valueddifferentialformsandtheexteriorcalculusispresentedinadedicatedlongchapterinafuturebookwhosecontentsisdescribedinthepresentchapter13.Forpresentpurposes,wearesimplyinterestedinmotivatingtheexteriorderivativeandprovidingtheconceptsneededtobefunctionalinthecaseofscalar-valueddifferentialforms,withoutprovidingproofofeverythingthatamathematicianwouldfindnecessitatingproof.Wehavefoundpertinenttotreatthesubjectofthischapterratherexpeditiouslyinordernottoblurthefocusonvector-valuednessofthisbook.Sincethenextonedealsalmostexclusivelywithscalar-valuedness,onewouldexpectthattheorderinthepublicationofthesebooksshouldbetheoppositeofwhatitisgoingtobe.However,whereasbook2willdealwithexterior-interiorcalculus,hereweonlydealwithexteriorcalculus.4.2DisguisedexteriorderivativeTheconceptofexteriorderivativeiscentraltotheCartancalculus,alsoknownasexteriorcalculus.ReaderswhoknowthetheoremsofGaussandStokesal-readyhavesomeknowledgeofexteriorderivativesandofanimportanttheoremaboutthesederivatives,butwithoutperhapsbeingawareofitall.Weshallintroducetheconceptfromtheperspectiveofthosewell-knowntheorems.Supposethatweaskourselveswhatshouldgoinsidetheintegralsontherightsideoftheincompleteequalitiesv(x,y,z)dy∧dz=?,(2.1)67

8668CHAPTER4.EXTERIORDIFFERENTIATIONv(x,y,z)dx=?.(2.2)Thedomainsofintegrationontheleftarerespectivelyaclosedsurfaceandaclosedlineinsimplyconnecteddomains.Theyaretheboundariesofthedomainsofintegrationoftheintegralsontheright.WecanusethetheoremsofGaussandStokeswith(vx,vy,vz)=[v(x,y,z),0,0]toanswerourquestion.Wethusobtain:∂v(x,y,z)v(x,y,z,)dy∧dz=dx∧dy∧dz(2.3)∂xand∂v∂vv(x,y,z)dx=dz∧dx−dx∧dy.(2.4)∂z∂yTheexteriorderivativeispreciselytheconceptthatpermitsustorespondtoourquestioninbothcases.Itisdefinedastheoperationthatmapstheintegralofanr−formextendedtoacloseddomainintotheintegralofan(r+1)-formextendedtothedomainSdelimitedbytheformerdomain,tobedenotedas∂S.Letωbeadifferentialr−form,understoodheretobetheintegrandofanr-integral(r=1,2,3...foraline,surface,volume,...).Fromaperspectiveofstructuralsimplicity,wewoulddefineitsexteriorderivativedωasthedifferential(r+1)-formthatsatisfiesω=dω,(2.5)∂SSassumingthatdωexistsindependentlyofS.Intheliterature,(2.5)isreferredtoasthegeneralizedStokestheoremorsimplyStokestheorem.Fromtheper-spectivejustmentioned,itisnotatheorembutanimplicitdefinition.Itisknownthatdωexists,butthatwouldbeatallordertoprove.Weshallproceedinadifferent,simplerway,likevirtuallyeveryauthoronthissubjectdoes.Butwewantedtoadvanceheretheperspectiveofthepreviousparagraph,sinceitbringstotheforewhatisgoingtobeachieved.Itisworthpointingoutthat,whereasdωisuniquelydefined,ω,ontheotherhand,isnotunique.Wemeantosaythatdifferentω’smayhavethesamedω,whichisreminiscentofthefactthatprimitivesinthecalculusofonevariablearedetermineduponlytoanadditiveconstant.Noticefromthisintroductiontotheconceptofexteriorderivativethatmoreappropriatenomenclatureforitwouldbethatofexteriordifferential,somethingtoberemembereduntilfamiliaritywiththeconceptpermitsreadersnottobebotheredbythisambiguityoftheterminology.Givenω,howdoesonefinddω?Equations(2.3)and(2.4)tellusthattheformsontheirrighthandsidesare,accordingtothecharacterizationjustgiven,theexteriorderivativesoftheformsonthelefthandsides.Morespecifically,Stokestheorem,viewedasimplicitdefinitionofexteriordifferentiation,permitsustofindthattheexteriorderivativeofvdxiis(∂vi/∂xj)dxj∧dxi.Similarly,iGausstheorem,againviewedasadefinition,permitsonetostatethatthe

874.3.THEEXTERIORDERIVATIVE69exteriorderivativeofvxdy∧dz+vydz∧dx+vxdx∧dyinthreedimensions,is(∂xvx+∂yvy+∂zvz)dx∧dy∧dz.Finally,theequationBf(B)−f(A)=fdxi(2.6),iAcanbeinterpretedtomeanthattheexteriorderivativeofascalarfunctionisjusttheordinarydifferentialdf=fdxi.(2.7),i(Inthiscase,whatisthedomainofintegrationonthelefthandsideof(2.6)?)Thosetheoremspermittedmathematicianstoinfertheformofdωforarbitraryformsω,onmanifoldsofarbitrarydimension.Oncetherulesforobtainingtheexteriorderivativewerefound,theargumentwasreversed.TherulesforobtainingexteriorderivativesbecametheirdefinitionandonethenprovedEq.(2.5)underthenameofStokestheorem.InSection3,weintroducethenewdefinitionand,inSection4,weprovethetheorem.4.3TheexteriorderivativeAnyr−formcanberepresentedasasumoftermsoftheformfdxi1∧...∧dxirwherefisazero-formorscalarvaluedfunction.Theexteriorderivativedofasumisdefinedasthesumoftheexteriorderivatives.So,whatremainsisthedefiningoftheexteriorderivativeofsimpleformsofranksoneandgreater.Givenτ=fdxi1∧...∧dxir,itsexteriorderivativeisdefinedas:d(fdxi1∧...∧dxir)≡df∧dxi1∧...∧dxir=fdxj∧dxi1∧...∧dxir.(3.1),jOnereadilyshowsthatddf=0sinceddf=d(fdxi)=fdxj∧dxi=(f−f)(dxj∧dxi),(3.2),i,i,j,i,j,j,iwhichiszerobecauseoftheequalityofthepartialderivatives.Inparticular,ddxi=0(Justfortherecord:Somemathematiciansmightdisagreewiththisapproachandstatethatddxi=0mustbeconsideredasapostulate;theymightberight).Noticethattheexteriorderivativesofformsofthehighestrank,n,arealsozero,thoughforadifferentreasonwhichshouldbeobvious.Onecanshowthatthedefinition(3.1)isindependentofcoordinatesystem,i.e.oneobtainsthesameresultbyperformingachangeofcoordinatesystemandthenexteriordifferentiatingaswhenoneexteriordifferentiatesandthenchangescoordinates.Wefinallydevelopaveryusefulexpressionfortheexteriorproductofformsofanyrank.Letusstartwiththeexteriorproductofjusttwoofthem.Thisproductcanalwaysbeexpressedasasumoftermsoftheformρ∧σwhere

8870CHAPTER4.EXTERIORDIFFERENTIATIONρ=Rdxi1∧...∧dxirandσ=Sdxj1∧...∧dxjs.Thend(ρ∧σ)=d(RS)∧dxi1∧...∧dxir∧dxj1∧...∧dxjs=(dR∧dxi1∧...∧dxir)∧(Sdxj1∧...∧dxjs)(3.3)+(−1)r(Rdxi1∧...∧dxir)∧(dS∧dxj1∧...∧dxjs)r=dρ∧σ+(−1)ρ∧dσ,wherethefactor(−1)risduetothefactthatwehavemovedthedS1−formfromthefronttotherightofdxi1...dxir.Byusingtheassociativepropertyoftheexteriorproduct,onegetsrr+sd(ρ∧σ∧τ)=dρ∧σ∧τ+(−1)ρ∧dσ∧τ+(−1)ρ∧σ∧dτ(3.4)andsoon.Animmediateconsequenceofddf=0isthatthesecondderivativeddρofanyr−formiszerosinceddxm=0,anddd(fdxi∧...∧dxj)=d(df∧dxi∧...∧dxj)=(ddf)∧dxi∧...∧dxi=0.(3.5)TheGaussand(traditional)StokestheoremsarebutparticularcasesofEq.(2.5),verificationofwhichweleavetoreaders.Again,ddf=0mayalsobeviewedasdefiningtheexteriorderivative;ddxi=0wouldthenbecontainedinthatdefinitionasparticularcases.4.4CoordinateindependentdefinitionofexteriorderivativeThedefinitionofexteriorderivativeofSection3.2isveryusefulforpracticalcalculations.Ithasthedisadvantage,however,thatitinvokesacoordinatesystemorpatch.Underachangeofcoordinatesfromxi’stoyj’s,thesimpler−formfdxi∧...∧dxjbecomesasumofr−formsfdyk∧...∧dym.Wek...mcannowapplytherulesofSection3.2tothislinearcombinationofr−formstoobtainitsexteriorderivative.Thisresultshouldcoincidewiththeresultoftransformingtotheycoordinatesystemtheexteriorderivativeobtainedinthecoordinatesystemx;abruteforceproofofthecoincidenceofthetworesultsisverycumbersome.Thisproblembecomesanon-issuebyintroducingacoordinateindependentdefinitionofexteriorderivative,whichwedointhefollowing.Givenanexteriorproductofformsα∧β∧...∧γofrespectivegradesa,b,...c,thedefinitiond(α∧β∧...∧γ)≡dα∧β∧...∧γ+(−1)aα∧db∧...∧γ(4.1)a+b+...+...+(−1)α∧β∧...∧dγapplies,inparticular,totheexteriorproductof1−forms,whichcanalwaysbeexpressedincoordinateindependentmanner.Wethushave:d(f0df1∧...∧dfr)=d[(f0df1)∧df2∧...∧dfr]=df0∧df1∧...∧dfr.(4.2)

894.5.STOKESTHEOREM71Thisdefinitionisexplicitlyindependentofcoordinatesystem.IttakestheformofEq.(3.1)whenthedifferentialformtobedifferentiatedisfirstwrittenasfdxi1∧...∧dxir.4.5StokestheoremInthissectionweshallproveStokestheorem,namelyEq.(2.5).Somereadersmayperhapswishtojustglanceatthissectionandproceedtothenextone.Weneedtofirstintroducetheconceptofpull-backofaform,orpull-backforshort.Thereaderhassurelycomeacrosspull-backsinthecalculuswithseveralvariables,wherepull-backsoccurwhenweperformlineandsurfaceintegrals.A(parametrized)surfaceinR3isamapU→R3whereU⊂R2.ThepointsofUwillberepresentedbythecoordinates(u,v)andthepointsR3bythecoordinates(x,y,z).Asanexample,theupperhemisphereHofunitradiusisamap(u,v)→(x,y,z)definedbyx=sinucosvy=sinusinvz=cosu,wherethedomainUis(0≤u≤π/2,0≤v<2π).Supposenowthatwewanttointegratetheformω=z2dx∧dyoverthehemisphereH,ω,ω=z2dx∧dyHThebestwaytoperformtheintegralistofirstobtaindxanddyintermsofduanddvandperformthesubstitutionsinω.Wethusobtain2ω=cosu(cosucosvdu−sinusinvdv)∧(cosusinvdu+sinucosvdv)=sinucos3udu∧dv(5.2)andfinally,ω=sinucos3udu∧dv.(5.3)HUTomakeapoint,wehavebeencarelesswiththenotationsince(5.2)and(5.3)togetherimplyω=ω,(5.4)HUwhichisnotcorrect.Theexpression(5.4)isincorrectbecauseωisnotdefinedonU.Theformz2dx∧dyisdefinedinR3,andwedenoteitasω.Theformsinucos3udu∧dvisdefinedinU⊂R2,andweshalldenoteitasS∗ω.Thisisasocalledpull-backoftheformω.Theconceptofpull-backcanbemademathematicallyrigorous.Theimportantthingtorealizeisthathereitsimplymeansthatwemovethedifferentialformfromonemanifoldtoanother.

9072CHAPTER4.EXTERIORDIFFERENTIATIONInsteadof(5.4),weshouldbestatingthatwearesayingthat∗ω=Sω.(5.5)HUIthappensthat,tostartwith,weshouldwrite∗∗2∗∗Sω=(Sz)(Sdx)∧(Sdy)S∗z=cos2u∗Sdx=cosucosvdu−sinusinvdvS∗dy=cosusinvdu+sinucosvdv.Inphysics,however,wewanttogettoresultsasfastaspossibledisregardingsometimestheuseofappropriatenotation,whichdistractsus.ForthederivationofStokestheorem,weneedmorerigorthatweareusedto.Letusformulatetheaboveconsiderationsinmathematicallanguage.LetAandMbetwodifferentiablemanifolds.LetUbeasubsetofAandconsideramapS:U→M.GivenafunctionfonM,wedefinefunctiongonUbymeansofg(Q)=f(P)whereQ∈UandwhereP=S(Q).The“pull-back”S∗ωoftheformω=fdf∧...∧dfonMisdefinedastheform01rgdg∧...∧dg.ThusthefunctionS∗transformsr−formsonMintor−forms01ronA.Onecanshowthat∗∗∗S(ω1∧ω2)=(Sω1)∧(Sω2)(5.6)andthatS∗(dω)=d(S∗ω).(5.7)WeproceedtoproveStokestheorem.Wewriteωasf0df1∧...∧df.Thusdωisdf0∧df1∧...∧dfr.LetSbeaparameterized(r+1)-surface,S:U→M,i.e.asmoothfunctionfromaclosed,boundedregionU⊂Rr+1.AssumealsothatUisconvex,i.e.containsalllinesegmentsbetweeneachpairofitspoints.Letα=S∗ω.Thendω=dα(5.8)SUandω=α(5.9)∂S∂Uwhere∂UistheboundaryofU.Ifweprovethatdα=α,(5.10)U∂Uweshallhaveprovedthatdω=ω.S∂SIf(u0,u1,...,ur)aretheparametersofS,wecanwriteαasα=adu1∧...∧dur+adu0∧du2∧...∧dur+...+adu0∧du1∧...∧dur−1.(5.11)01r

914.6.DIFFERENTIALOPERATORSINLANGUAGEOFFORMS73Weshallprovethatd(adu1∧...∧dur)=adu1∧...∧dur.(5.12)00U∂U(Theothertermsaredoneanalogously).Iftheboundaryisconvex,ithasanupperandalowerpart,respectivelyu0=f(u1,...,ur)andu0=f(u1,...,ur)21withf≤f.Wethushave,withdu1...r≡du1∧...∧dur,12f21...r∂a001...rf21...rd(a0du)=∂u0du∧du=[a0]f1duUf1=[a(f,u1,...,ur)−a(f,u1...r)]du1...r(5.13)0201=adu1...r.0∂UQ.E.D.ReaderswillhavenoticedthattheproofisessentiallythesameasfortheusualStokestheoreminvolvingtwo-dimensionalsurfacesin3-DEuclideanspace.Wehaveleftoutsomemattersrelatedtoorientationoftheforms,whicharenotunliketheproblemoforientationoftheboundingcurveintheusualStokestheorem.Supposefinallythatthereweresomeotherdifferentialformμ,i.e.otherthandω,thatsatisfiedω=μ.(5.14)∂SSWewouldthenhavedω=μ(5.15)SSforanyapplicableintegrationdomain.Sincethesecanbeassmallaspossible,thetwodifferentialformsdωandαwouldhavethesamecoefficientsatanypoint.Theywouldthusbethesamedifferentialform.Inotherwords,theexteriorderivativeisuniquelydefined.4.6DifferentialoperatorsinlanguageofformsOurthree-spaceofeverydaylife(three-dimensionalEuclideanspace)isaveryspecialcaseofdifferentiablemanifold;ithasmuchadditionalstructure(Chap-ters6).Ifthisspacewerenotendowedwithadotproduct,theconceptsofgradient,curl,divergenceandLaplacian(divergenceofgradient)couldnotbedefinedwithinthevectorcalculus,whichisoneofitsdeficiencies.Gradientandcurlshouldnotrequiresuchaproduct,thusasocalledmetricstructure.Ourcomputationswillmakethatclear.Thedivergencedoesbut,fortunately,onecouldeasilypretendthatitdoesnotinEuclideanspaces.Weshalldefinethoseoperationswiththerolesofvectorfieldsreplacedbydifferentialforms.

9274CHAPTER4.EXTERIORDIFFERENTIATIONLetMbeann−dimensionaldifferentiablemanifold.Theexteriorderivativeofascalarfunction(zeroform)willbecalleditsgradientdifferentialformdf=fdxi.(6.1),iLetnowαbea1−form,α≡adxi.Itsexteriorderivativeiijijidα=d(aidx)=ai,jdx∧dx=(ai,j−aj,i)(dx∧dx)(6.2)willbecalleditscurl.Thesedefinitionsarevalidonanydifferentiablemanifoldand,sincetheyareinvariants,inanycoordinatesystem.Itisnowamatterofspecializingtowhateverbasiswewishtouse.DivergencesandLaplaciansareadifferentstory,asthemetricstructureisthenessential.Theirdetailtreatmentwilltakeplacewhenweshallintroducethatstructureinalaterchapter.Fortunately,wedonotneedtowaituntilthentopresentinEuclidean3-Dashortcuttotheconceptofdivergence.Givenavectorfieldb≡b1i+b2j+b3k(6.3)in3-DEuclideanspace,weassociatewithitinCartesiancoordinatesthedif-ferential2−formβ=b1dy∧dz+b2dy∧dz+b3dx∧dy(6.4)butonlyinCartesiancoordinates(ittakesotherformsinothersystemsofcoordinates,asweshalllatersee).Roughly,thedivergenceofthevectorfieldbcanbereplacedwiththeexteriorderivativeofβ,∂b1∂b2∂b3idβ=++dx∧dy∧dz=b,idx∧dy∧dz,(6.5)∂x∂y∂ztogreatadvantage(Indeed,readerswillidentifythecontentsofthesquarebracketasthedivergenceofb1i+b2j+b3kinCartesiancoordinates.Divergencesareidentifiedwithdensities,whicharetobeintegrated.Therighthandsideof(6.5)isareadytouseintegrand.Wemightwishtorelatethegradientandcurldifferentialformstothegra-dientandcurlvectorfields.Inthevectorcalculus,thegradientisgradf=fei.(6.6),iNoticethatthebasishastogowithsuperscriptstoindicateitsappropriatetransformationpropertiessothatthegradientwillbeaninvariant.Theprob-lemis:whatisei?Givenacoordinatesysteminadifferentiablemanifold,acorrespondingbasisvectorfieldeiisdefinedevenifthetangentvectorspacestothemanifoldarenotEuclidean(absenceofmetricstructure).Adotproduct,equivalentlymetricstructure,isneededinadifferentiablemanifoldinordertodefineei,aswesawinthepreviouschapter.Butwhyshouldweinvokestructurethatisnotneededinordertoachievethesameobjectives?

934.6.DIFFERENTIALOPERATORSINLANGUAGEOFFORMS75ForthemomentletusrecallwhateveryreaderknowsaboutmetricsinEuclideanspaces.Theyaretheexpressionsds2≡gdxidxj,(6.7)ijexamplesbeingin3-DEuclideanspaceds2=dr2+r2dθ2+r2sin2θdφ2(6.8)insphericalcoordinatesandds2=dρ2+ρ2sin2θdφ2+dz2(6.9)incylindricalcoordinates.Orthogonalcoordinatesystemsarethosewherethemetric(tobedefinedinchapter6forEuclideanspaceandinchapters9forgeneralizationsofEuclideanspaces)takestheformi=nds2≡(ωi)2,(6.10)i=1withωi≡hi(x)dxi(nosum).(6.11)Everymetriccanbewrittenas(6.10),inaninfinitenumberofways,sinceoncewehavefoundonesolution,wecanfindothersolutionsbyapplying,inEuclidean3-D,arbitraryrotations.Butnoteverygdxidxjadmitsωi’softheijtype(6.11).Orthogonalcoordinatesystemsdo,andareeasilyrecognizablesinceds2thentaketheformds2=(h1dx1)2+(h2dx2)2+(h3dx3)2+...,(6.12)whichembodies(6.10)and(6.11).Electricalengineersandcertaintypesofphysicistswillneedtohavethedifferentialoperatorsintermsoforthonormalvectorbasisfields.Thegradientvectorfielddoesnottaketheform(6.6)ingeneralorthonormalframefields.Computingintermsoforthonormalbasisfieldsisequivalenttocomputingintermsofbasesωi’sofdifferentialformsthatsatisfy(6.10).But(6.11)willnotbesatisfiedingeneral.wenaturallyhaveiidf=f,idx≡f/iω.(6.13)Forourpresentpurposes,onemaythinkofthef/iasiftheywerethecompo-nentsofavectorfieldintermsofanorthonormalbasisfield.Wenowspecializethecoordinateandbasisindependentequations(6.1)-(6.2)andtheCartesianspecificequation(6.5)toorthonormalvectorbasisfieldsassociatedwiththesphericalandcylindricalcoordinates.Itisobviousfrom(6.11)and(6.13)thatwecanrewrite(6.1)asdf≡[(hi)−1f](hidxi)=[(hi)−1f]ωi.(6.14),i,i

9476CHAPTER4.EXTERIORDIFFERENTIATIONThecomponentsofdfrelativetoωiwillbefamiliartoengineersandphysi-cistsusingthevectorcalculuswithorthogonalcoordinatessystems,likethecylindricalandsphericalones.Somereadersmaywonderwhetherwehavelosttheinformationprovidedbythegradientvectorfield,namelythedirectionofmaximumchangeofascalarquantityandthemagnitudeofsuchachangeperunitlength.Onemayspeakofchangeperunitlengthonlyiflengthisdefinedinthefirstplace,whichametricdoes.Hencethegradientdifferentialformdoesnotprovidethatinformationifthedifferentiablemanifoldisnotendowedwithmetricstructure;butthegradientvectorfieldisnotevendefined.Ifametricisgiven,wemaydefineavectorfieldwithcomponentsgivenbythesquarebracketsin(6.14).Considernowthecurl,i.e.(6.2).TheFj’sdefinedbyα≡adxj≡Fωj=Fhjdxj(6.15)jjjwillplaytheroleofthecomponentsofavectorfieldintermsofareciprocalbasis.Hence,from(6.15),wegetjaj=Fjh(nosum),(6.16)whichwetaketo(6.2)toobtaindα=[(Fhi)−(Fhj)](dxj∧dxi)i,jj,i(6.17)=(1/hh)[(Fhi)−(Fhj)](ωj∧ωi).iji,jj,iEndofstory.Thecomponentsofthis2−formarethesameonesasthosethatgobythenamecurlvectorfieldintheliteraturei/h2h3j/h1h3k/h1h2∂/∂x1∂/∂x2∂/∂x3,(6.18)Fh1Fh2Fh3123intermsoforthonormalbasisfieldsassociatedwithorthogonalcoordinatesys-tems(WehaveusedFi=F).Thematchingofcomponentsaofadifferentialiij2−formandcomponentsofavectorfieldisapeculiarityofthreedimensions.Themetricisused(wedonotsayhowatthispoint)toassociatethedifferential2−formwiththevectorfield,nottodefineaconceptofcurldifferentialform.Thecaseofthedivergenceisdifferent.Herethemetricisessentialevenindealingwithdifferentialforms.Ithastodowithsomethingwhich,inK¨ahler’sgeneralizationoftheexteriorcalculusofdifferentialforms,goesbythenameofinteriorderivative.Therearestructures,likeEuclideanspaces,wherethedivergenceofanobjectofgradercanbereplacedwiththeexteriorderivativeofanassociateddifferentialformofgraden−r.Recallthatwewentfromthevectorfield(6.3),whichisofgradeone,tothedifferentialform(6.4),whichisofgrade3-1.Correspondingtoorthonormalbases,wedefinenewcomponentsofβbyβ=F1ω2∧ω3+F2ω3∧ω1+F3ω1∧ω2.(6.19)

954.7.THECONSERVATIONLAWFORSCALAR-VALUEDNESS77Clearlybi=Fihjhk,withthethreeindicesmakingacyclicpermutationof(1,2,3).In(6.5),wereplacetheseexpressionforbianduse(6.11)toget∂(F1h2h3)∂(F2h1h2)∂(F3h1H2)dβ=++dx1∧dx2∧dx3∂x1∂x2∂x3(6.20)1∂(F1h2h3)∂(F2h1h3)∂(F3h1h2)=++ω1∧ω2∧ω3.h1h2h3∂x1∂x2∂x3Readerswillrecognizethecoefficientofthis3−formasthedivergenceofavectorfieldofcomponentsFiintermsofannormalizedbasisfielddualtoanorthogonalsystemofcoordinates.4.7Theconservationlawforscalar-valuednessAdirectconsequenceofStokestheoremisthefactthat,iftheexteriordifferentialdαrofanexteriorr−formαriszeroonasimplyconnectedregionofamanifold,wehaveαr=0(7.1)∂Ronanyclosedr-surface∂Rinthatregion.Equation(7.1)isthestatementoftheconservationlawofdifferentialforms.Weshallseebelowthatthislawcomprises(familiarformsof)morespecializedconservationlaws.Thespecial-izationtakesplacesometimesbysplitting∂Rintopieces,usuallytwo(likethetwosemispheresofasphere)orthree(likethetwobasesandthelateralfaceofacylinderinspacetime).Theconservationlawinthecaseofscalar-valueddifferentialformsisthegeneralizationtothoseobjectsofthestatementthat,ifdfisnull,fisaconstant.Equation(7.1)doesnotexplicitlysaythat,ifdαriszero,αrisaconstantdifferentialform.Wehavenotevendefinedwhat“constantdifferentialform”couldpossiblymean.Theconstancyoffifdf=0isapeculiarityofdαbeinga1−form,i.e.afunctionofcurves.αisthenafunctionofpoints.Moreprecisely,theboundaryofanopencurveisapairofpoints.Evaluatingaconstantasifitwereazero-dimensionalintegrandistheninterpretedtomeanf(B)−f(A)=0.(7.2)Thisappliesto“anycurvewithinourcurve”(i.e.toanypairofitspoints)andtoanycurvebetweenanytwopointsonsimplyconnectedregionsofmanifold.Theconstancyofffollows.ThereistheissueofwhytheminussigninEq.(7.2).Thishastodowithorientation,aconstantthemeintheexamplesthatfollow.Asweindicated,weshallsplit∂Rintopieces.Howwebreakitdependsontherankofthedifferentialform,especiallyinrelationtothedimensionalityofthemanifold(thus,forexample,onwhetherweconsidertheelectricfieldasapieceofadifferential2−forminspacetimeorasadifferential1−formin

9678CHAPTER4.EXTERIORDIFFERENTIATION3-space).Thesignaturealsomatters.Forinstance,ifweconsider3-Dcylindersinspacetime,wetakethemwithaxisalongthetimedimension.Considerthedifferential1−formEdxi.Ifd(Edxi)iszero,StokestheoremiiimpliesthatEdxiiszeroonclosedcurves.LetAandBbetwopointson∂Rionesuchcurve.Noticehowaminussignappears:BABB0=Edxi=Edxi+Edxi=Edxi−Edxi,(7.3)iiiii∂RABA;(1)A;(2)wherewehaveused(1)and(2)todenotetwodifferentpathsbetweenAandB.Considernextthe2−formBdxj∧dxk,wherewesumovercyclicpermuta-itionsof(1,2,3).Assumethatd(Bdxj∧dxk)iszero.TheniBdxj∧dxk=0.(7.4)i∂RWedivide∂Rintotwoopensurfaces:∂R=⊕12.Then0=Bdxj∧dxk+Bdxj∧dxk.(7.5)ii1;out2;outSupposethatthetwosurfaces1and2,whichareboundedbythesamecurve,wereveryclosetoeachother.Itisclearthat“outinonecase”meanstheoppositeof“outintheothercase.”Hence,forcommonorientation,theremustbeanegativesignbetweenthetwointegrandsontheright,andtheirequalityfollows.TheintegrationoftheBdifferentialformisthesameforallopensurfaceswithacommonboundaryprovidedthattheyareconsideredallatthesameinstantoftimeandequallyoriented.ThiscaveatmeansthattheBfieldistobeconsideredasadifferential2−form.EandBtogetherconstitutethedifferentialformF=Edt∧dxi−Bdxj∧dxk,(7.6)iiwhichhastheconservationlaw0=Edt∧dxi−Bdxi∧dxk(7.7)ii∂Rassociatedwithit,where∂Ristheboundaryofa3-surface,andis,therefore,aclosedspacetimesurface.Inspacetime,considera∂Ratconstanttime,i.e.apurelyspatialclosedsurface.ThefluxofBover∂Riszero.Inordertostudytheevolutionwithtimeofintegralsatconstanttime,weintegrateonarectilinearcylinderwithequalspatialbasesattimest1andt2.Wehavesixtermstotakecareof:theintegrationofbothEdx0∧dxiandof−Bdxj∧dxkoverthetwobasesandiioverthelateralsurface.TheE-integralsoverthebasesandtheB-integraloverthelateralsurfacearezero.Onethusobtainst20=dtEdxi+Bdxjdxk+Bdxjdxk.(7.8)iiit1outatt1outatt2

974.8.LIEGROUPSANDTHEIRLIEALGEBRAS79Onedifferentiateswithrespecttot2atconstantt1,andobtainsFaraday’slaw.Thatiswhathavingaconserved2−forminspacetimemeansintermsofquan-titiesatconstanttime.Theotherconservationlawofelectrodynamicsinvolvesthecurrent3−form:j=jdx1∧dx2∧dx3+jdt∧dx2∧dx3+jdt∧dx3∧dx1123023031(7.9)+jdt∧dx1∧dx2.012Theconservationlawthatisaconsequenceofdj=0readsj=0.(7.10)∂RThej123componentisthechargedensity,ρ.Wetakeas∂Rtheboundaryofa4-dimensionalcylinderwhoselateral“surface”(three-dimensional!)ispushedtoinfinity,withbasesatconstanttime.Theresultistheconservationofcharge.Itisseldommentionedthatonedoesnotneedtointegrateatconstanttimetoobtaintheamountofcharge.Onealsoobtainsitifoneintegratesjoveranysectionofthe4-dimensionaltubeofworldlinesofallchargesinthevolume.Hencethecomponentsj0lm(equivalently,thecomponentsofwhatisconsideredavectorfieldjintheliterature)areinstrumentalnotonlyinobtainingthefluxofchargethroughasurfacebutalsoincomputingthetotalchangeusingasectionnotatconstanttime(see“Le¸conssurlesinvariantsintegraux”byE.´Cartan[9]).Aftertheexperiencegainedwithorientation,wereturntowhenαisa0−form.Theorientedboundary∂Rnowisthepairofpoints(A,B)togetherwith“directionawayfromthecurve.”Directionawayfromthecurveassignsoppositesignstotheevaluationoffatthetwoends,asbecomesobviousbymakingthecurvesmallerandsmalleruntilitreducestoapoint.OnceagainifRisacurve,0=df=f=f(∂R)=f(B)−f(A),(7.11)R∂Rwhichthusisseenasonemoreexampleofapplicationof(7.1).4.8LieGroupsandtheirLiealgebrasThetitleofthissectionismeanttode-emphasizetheratherabstractconceptofLiealgebraandfocusinsteadontheeasiertounderstandconceptofLiealgebraofaLiegroup,whichisofinterestinthisbook.MostphysicistswillavoidsubtletieswhenreferringtoaLiegroupandde-fineitasjustacontinuousgroup.Amathematicianwillrightlysaythatthisisnotgoodenough,fortworeasons.First,wewanttohavemorethancon-tinuity,namelydifferentiability,andthatiswhatdifferentiablemanifoldsarefor.ThusaLiegroup,G,isagroupthatisalsoadifferentiablemanifold.Asecondrequirementisacompatibilityconditionbetweenthetwostructuresof

9880CHAPTER4.EXTERIORDIFFERENTIATIONgroupanddifferentiablemanifold.Butweshallnotenterintothatsincetheconceptissatisfiedinthetheorythatweshallbedeveloping.Ittakesaskilledmathematiciantothinkofexampleswheretheconditionisnotsatisfied.Although,aswesaid,wearenotinterestedinLiealgebrasingeneral,letusstatethattheyaremodulesinwhichwehaveorcanintroduceamultiplicationofatypecalledLieproduct.Ifwekepton,wewouldbespeakingofLiebrackets,butthiswouldtakeusintounnecessaryanddistractingconsiderationsonvectorfieldsthatwewishtoavoid.(Readerswhomighthavedifficultyinthefollowingwithouruseoftermslikeplanesinthepresentcontextshouldjumpnowtotheopeninglinesofsection7.2,andcomeback.)Letgdenotethegeneralelementofagroupoftransformations.Letdgbeitsexteriorderivative.Letui,(i=1,...,m),beacompletesetofparametersinthegroup,equivalentlyacoordinatesysteminthemanifold.dgisalinearcombinationdg=g,dui≡fdui.(8.1)iiIfGwereagroupofmatrices,thefduiwouldbematriceswithentriesthatiaredifferential1-forms.Let(Δui)denotem−tuplesofrealnumbers.UsuallyGwillbeapropersubgroupofthegroupofregularmatricesn×n.ThesetofmatricesfΔuithenisanm-dimensionalhyperplaneinthemanifoldofsuchin×nmatrices.OnemightthinkthatfΔuiisthetangentplaneatg.Itisnotiquitethat;itsimplyisparalleltothattangentplane.Thiswillbecomeeasiertounderstandinthefollowing.Considerthedifferentialformω≡(dg)g−1=(fdui)g−1=(fg−1)dui,(8.2)iicalledMaurer-Cartanformofthegroup.Correspondingly,weshallhaveaplane(fg−1)Δui.(8.3)iTheunitelementeofthegroupisapointofthemanifold.Similarly,anyotherelementofthegroupisapointofthemanifold.Onecanmakethefollowingstatement:ginthegrouptakeseinthemanifoldtoginthemanifold.So,g−1takesg,itsneighborhoodanditstangentplanerespectivelytoe,itsneigh-borhoodanditstangentplane.Hence(fg−1)Δuiisindependentofg.Whatisiit?(fg−1)ΔuiwouldbethetangentplaneateiffΔuiwerethetangentplaneiiatg.Though(fg−1)ΔuiandfΔuilooklikesuchtangentplanes,theyarenotiiquiteso.Hereiswhy.(fg−1)Δuiande+(fg−1)Δuiareparallelplanes.So,iitheydonothaveanypointincommon.Sinceeisine+(fg−1)Δui,itcannotibein(fg−1)Δui.Clearly,(fg−1)Δuiistheplaneatthenullmatrixthatisiiparalleltothetangentplaneattheunitmatrix,i.e.attheunitelement.TheMaurer-CartanformofthegroupsatisfiesthesocalledMaurer-Cartanequationofstructure:dω=ω∧ω.(8.4)

994.8.LIEGROUPSANDTHEIRLIEALGEBRAS81Weshallhavetheopportunityofseeingthisequationemergenaturallyinseveralcasesofinterestforus.Hence,wedonotinsistonthisatthispoint.Inordertohaveanalgebra,weneedasumandaproduct.Thesumisthesumofmatrices.Buttheproductisnottheproductofmatrices,buttheiranti-symmetrizedproduct.Thisagain,weshallseelaterindetail.Letusnowpracticealittlebitwithsomeverysimpleω’sandcorrespondingLiealgebras.Considerthematricesfortheonedimensionalgroupofrotations.Wehave−1−sinφ−cosφcosφsinφ0−1ω≡(dg)g=dφ=dφ,cosφ−sinφ−sinφcosφ10(8.5)brieflywrittenasω=adφ,with0−1a≡.(8.6)10Considernowthegroupoftranslationsx=x+u(8.7)inonedimension.Thisisnotalineartransformation,sincetranslationsindimensionnarenotgivenbymatricesinthesamenumberofdimensions.n-translationscan,however,berepresentedbymatricesinn+1dimensions.Thus(8.7)canberepresentedasx1uxx+u==.(8.8)10111ωisthengivenby0du1−u01ω==du=bdu,(8.9)000100with01b≡.(8.10)00Thegroupofrotationsiscompactandthegroupoftranslationsisnoncompactbut,inonedimension,theyarelocallyisomorphic,whichistosaythattheirLiealgebrasareisomorphic.Thecompositionlawinthealgebrais(abstractionmadeofaandb),Δφ1+Δφ2andΔu1+Δu2respectively.Whenoneknowsthatthereisabetteroption,onerealizesbycomparisonthatmatricesarecumbersometoworkwith.InthecaseofrotationsandofLorentztransformations,thereisamuchbettermathematicaltoolthanmatrixalgebra.ItisCliffordalgebra.

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103Chapter5AFFINEKLEINGEOMETRYWeleaveforchapter10theexplanationofwhyweusethetermKleingeometryinthetitleofthispartofthebook.WeshallthenhavetheperspectiveofhavingdealtwithgeneralizationsofKleingeometriesthatremain“Klein”insomesense,andshallobtaintheperspectiveofgeneralizationsthatceasetobeKleineveninthatlimitedsense.Forthemomentandtokeepgoing,Kleinmeansflat.Betteryet,affineKleingeometryandEuclideanKleingeometrysimplymeanaffineandEuclideangeometry.5.1AffineSpaceTheconceptofaffineconnectionisofutmostimportanceindifferentialgeometry.ItderivesfromtheconceptofaffinespaceAfn.Geometryinaffinespaceiscalledaffinegeometry.InitsKleineanstage,affinegeometryistechnicallydefinedasthepair(G,G0)constitutedbytheaffinegroup,G,anditslinearsubgroup,G0.Inthispresentation,thesegroupswillemergefromtheconceptofaffinespace.AratherintuitivedefinitionofAfnisthefollowing:asetofpoints(A,B,...)suchthattoeachorderedpairofpoints(A,B)wecanassociateavectorAB∈nVwiththefollowingproperties:(a)AB=−BA,∀A,B∈Afn;(b)AB=AC+CB,∀A,B,C∈Afn;(c)GivenanarbitrarypointAandvectorv,apointBexistssuchthatAB=v.Thevectorspaceiscalledtheassociatedvectorspace.Inthisbook,theassociatedvectorspaceswillbevectorspacesoverthereals.Toputitslightlydifferently,affinespaceisapairofavectorspaceandanassociatedsetofpoints85

10486CHAPTER5.AFFINEKLEINGEOMETRYrelatedtoitintheaforementionedway;avectorvactingonapointAyieldsapointB.LetusmakeA=B=Cinthedefiningproperty(b).ItimpliesthatAAisthezerovector.Butthisalsoappliestoanyotherpoint.Hence,inprinciple,anypointcanbeassociatedwiththezerovector.Thereisaonetoonecorrespondencebetweenthepointsofaffinespaceandthevectorsofavectorspaceonlyafteroneparticularpointhasbeenarbitrarilyassociatedwiththezerovector.LetuscallthatpointO.ApointQcanactuallybegivenasQ(=OQ).Fromnowon,wesimplyeliminatethepointOfromthenotation.AreferencesysteminaffinespaceisconstitutedbyapointQandavectorbasisatQoftheassociatedvectorspace,whichisequivalenttosayingthatwehaveabasis(ai)atQ.Withtheseconventions,wecanwriteforarbitrarypointPandspecificallychosenpointQP=Q+Aia.(1.1)iIngeneral,givenacoordinatesystem(x)(alsocalledachart),thecoordinatelinexiisdefinedasthelocusofallpointssuchthatxjisconstantforallvaluesofjexcepti(j=1...i−1,i+1...).Inparticular,thecoordinatelineAihasthedirectionofthebasisvectorai,i.e.,thevectorassociatedwithanypairofpointsinthislineisproportionaltothebasisvectora.TheAi’sarereferreditoasrectilinearcoordinatessincetheycorrespondtostraightlines.InFigure1,thepointsPandShavethesamecoordinateA2;thepointsPandRhavethesamecoordinateA1.ThevectorsRandSarerespectivelyequaltoA1aandA2a.Therectilinearcoordinatesarethereforeassociatedwiththe12parallelprojectionsofthevectorPupontheaxes,i.e.uponthecoordinatelinesgoingthroughtheoriginQ.a22SPAa2QaR11Aa1Figure2:EmergenceofrectilinearcoordinatesConsideranewframe(Q,a,a)atthesamepointQofAfn.Intermsof1...nthenewbasis(a),wecanexpandPasiP=Q+Aia.(1.2)iTherelationbetweentheAandAcoordinatesareobtainedfromappropriatesubstitutionin(1.1)or(1.2)oftherelationbetweenthebases.

1055.2.THEFRAMEBUNDLEOFAFFINESPACE875.2TheframebundleofaffinespaceTheframebundleFnofAfnisthesetofallitsframes.Itsnamereflectsitsbeingendowedwiththesophisticatedstructurethatwillbethesubjectofthissection.Giventheparticularframe(Q,aa),anyotherframe(P,a,a)1...n1...ncanbereferredtothisonethroughtheequationsP=Q+Aia,a=Ajiiiaj.(2.1)Thereisaone-to-onecorrespondencebetweentheaffineframesandthesetof2jjn+nquantities(Ai,Ai)withdet[Ai]=0.Readersfamiliarbynowwiththeconceptofdifferentiablemanifoldwilleasilyverifythattheframebundleconstitutesadifferentiablemanifoldofdimensionn+n2.Otherreadersneedonlybelievethatthisamanifoldwherewecandifferentiatewithoutworry.Thepointsoftheframebundlemanifoldarethustheaffineframes.Thequantitiesjj(Ai,Ai)withdet[Ai]=0constituteasystemofcoordinatesontheframebundlemanifold.Inotherwords,theyassignlabelstoframes.Athirdframe(R,a,...,a)isgivenintermsof(P,a,...,a)by1n1nR=P+Aia,a=Aj.(2.2)iiiajSubstitutionof(2.1)in(2.2)yields(R,a,...,a)intermsof(Q,a,...,a).1n1nTheresultingexpressionremainsofthesameform.Wemayviewthesepairsofequationsastransformations.Theyconstituteagroup.(Readers,pleasecheckthattheconditionsdefiningtheconceptofgrouparesatisfied.)ItiscalledtheaffinegroupAf(n).BychoosingaframeinFnaspreferred,wehavebeenabletoputthemembersofFninaone-to-onecorrespondencewiththemembersofAf(n).Theparameters(Ai,Aj)inthegroupareasystemofcoordinatesintheiframebundle.Inthefollowing,weshallcontinuetodenoteafixedframeas(Q,a1,...,an),andarbitraryframes(saidbetter,frame-valuedfunctions)as(P,e1,...en).Thus,weshallwrite:ijP=Q+Aai,ei=Aiaj,(2.3)fortheactionoftheaffinegroup.Thesecondoftheseequationsrepresentstheactionofitsmaximallinearsubgroup.Theelementsofthematrixinversetojj[Ai]willbedenotedasBi.Aformalizationoftheseconceptsnowfollows.ThenaturalmapP:Fn→Afn(i.e.(P,e...e)goestoP)isacontinuousnandontoprojection.Wedefinesimilarprojectionmapse.AfniscalledtheibaseoftheframebundleFn.ThesetofallpointsoftheframebundlethataremappedtoPiscalledthefiberaboveP.ThefibersofFnaren2-dimensionalmanifoldswhosepointsareinaone-to-onecorrespondencewiththeelementsofthegenerallineargroupGL(n)(i.e.ofalllineartransformationsinndimen-sions),onceaparticularbasisateachpointPhasbeenchosenas“preferred”.jThecoordinatesinthefibersaretheAi’s.Wecanvisualizeframebundlesasbundlesoffibers.RefertoFigure2.Anamelikefiberbundleismorepictorialbutlessprecisethanframebundle.A

10688CHAPTER5.AFFINEKLEINGEOMETRYframebundleisafiberbundlewherethefibersaremadeofvectorbases;thepointtheyareatmakesthebasesbecomeframes.Buttherearealsofibersmadeofvectors,spinors,etc.Theyarethencalledvectorbundles,spinorbundles,etc.Veryprecisedefinitionsoffiberbundles,principalfiberbundles,framebundles,tangentbundles,etc.existintheliteratureondifferentialgeometry.Wecandowithouttheirformalintroduction.nFGL(n)GL(n)PnAfFigure3:FibrationofFnoverAfnOwingtothefactthattheframe(Q,a1,...,an)isfixed,wecanreadilydifferentiatethevectorvaluedfunctions(2.3)asfollows:ijdP=dAai,dei=dAiaj.(2.4)(somemayprefertothinkof−Pasthevectorfield—definedbyafixedpointQ—thatassignstoeachpointPthevectorfromPtoQ).ThedAi’sanddAj’sspann−dimensionalandn2-dimensionalmodulesofi1−forms.WecanthinkofdPasdA1⊗a+dA2⊗a+...andofdeas12idA1⊗a+dA2⊗a+...OnethusreferstodPanddeasvector-valuedi1i2idifferential1−forms.Oneshouldkeepinmind,however,thatdeiisnotvector-valuedinthesamesensethatdPis.Theboldfacedquantitieshavesubscriptsinonecase,butnotintheother.Thatisaveryimportantdifferencewhoseconsequenceswillbecomeincreasinglyapparent.Thepresenceoftheconstantbasis(ai)inEqs.(2.4)isunnecessaryandhidesthequalityofinvarianceoftheconceptswithwhichwearedealing.Weproceedtoremovethatbasis.Solvingforaiinthesecondequation(2.4)wehavejai=Biej.(2.5)Substituting(2.5)in(2.4),weobtaindP=dAjBie,de=dAkBje,(2.6)jiiikjwhichstatesthatdPanddeiarelinearcombinationsoftheei’swithcoeffi-cientsdAjBianddAkBjthattogetherconstituten+n2differential1−formsjik

1075.3.THESTRUCTUREOFAFFINESPACE89inthebundle.Wedenotethesecoefficientsasωiandωj.Intermsoftheiaforementionedcoordinatesystem,theyaregivenbyωi=dAjBi,ωj=dAkBj(2.7)jiikandarereferredtoastheleftinvariantformsoftheframebundleofAfn.Ingeneral,theywilllookdifferentintermsofothercoordinatesystems(Wespeakofleftand/orrightinvariantsdependingonwhetherwemakegroupsactontheleftorontheright;inthisbook,itwillalwaysbeleftinvariants).Assumewearegivenasetofn+n2independentdifferentialformsthatcannotbereducedtotheform(2.7)byachangeofcoordinates.Inthatcasetheyrepresentastructureotherthananaffinespace.Iftheystillsatisfycertainconditionstobelaterconsidered,theymaybeusedastheωiandωjintheisystemijdP=ωei,dei=ωiej,(2.8)towhichCartanreferredtoastheconnectionequations.Theseformsarethencalledtheconnectiondifferentialforms.Inthenextsection,weshallconsiderconditionstobesatisfiedbyaset(ωi,ωj)tobereducibletotheform(dAj,dAk)byasetofcoordinatetrans-iiformations.Ifitisreducible,theframebundledefinedbythe(ωi,ωj)willbeitheoneforanaffinespace,andthecoordinatetransformationwilltakeusfromcurvilineartorectilinearcoordinates.Theidentificationoftheactualtransfor-mationisanadditionalproblem.Sufficetosaythat,iftheconditionsaresat-isfied,wemayintegratethe(pull-backofthe)differentialforms(ωi,ωj)alongianycurvebetweentwopointsandrelateanytwopointsoftheframebundleinapath-independentway.When,goingtheotherway,wechangefromtherectilinearcoordinatesAitocurvilinearcoordinatesxi,theωjwillalsodependonthexi.If,inaddition,kwechangethetangentbasisfieldintermsofwhichweconstructthebundletojanon-constantbasisfield,asweshalldolater,theω’swillalsodependontheidifferentialsofthosecurvilinearcoordinates.Fortheconceptofconstantbasisfield,recallforthemomentwhatwassaidinsection6ofchapter3.Wereturntothisinsection4.Theconnectiondifferentialformsaresaidtobeinvariantbecausetheyareindependentofcoordinatesystem,evenifoneusescoordinatestorepresentthem.Thisandmanyotherstatementsmadesofarwillbeclarifiedwiththeintroductionofothercoordinatesystemsinsection5,andwiththesimplerbutstructurallysimilarexamplesofchapter6,andalsowiththesimpleexamplesofCartaniangeneralizationsofchapter7.5.3ThestructureofaffinespaceThesystemofequations(2.4)isobviouslyintegrabletoyield(2.3).Thesystem(2.6)alsoisintegrable,sinceitisadifferentversionof(2.4),andtheresultoftheintegrationagainis(2.3).Considernextthesystemofdifferentialequations

10890CHAPTER5.AFFINEKLEINGEOMETRY(2.8)whereωiandωjdonottaketheform(2.7).Theymay—ormaynot—beithesamedifferentialforms(2.7),thoughexpressedintermsofsomeothercoor-dinatesystem.Iftheyare(arenot),wesaythatthesystem2.8is(respectivelyisnot)integrable.TheissueofintegrabilityisresolvedbyapplicationofatheoremcalledFrobe-niustheorem.Itsapplicationtothesystembeingconsideredhereresultsinthefollowing:anecessaryandsufficientconditionfortheintegrabilityofthesys-tem(2.8)(withratherarbitrarydifferentialformsωiandωj;moreonallthisitobefoundinalaterchapter)isthatd(ωie)=0,d(ωje)=0.Thenecessityiijconditionisobvious.Sufficetodifferentiatetherighthandsideof(2.4):ddP=d(dAia)=ddAia+dAida=0+0=0,(3.1)iiijjjddei=d(dAiaj)=ddAiaj+dAidaj=0+0=0.(3.2)Weshallnotprovethesufficiencycondition.LetuswriteddP=0andddei=0intermsoftheinvariantformsofthebundle.Differentiating(2.8),weget:0=dωie−ωj∧de=(dωi−ωj∧ωi)e,(3.3)ijji0=dωje−ωk∧de=(dωj−ωk∧ωj)e.(3.4)ijikiikjHence,sincethe(ei)isavectorbasis,wehave:dωi−ωj∧ωi=0,dωj−ωk∧ωj=0.(3.5)jiikThesearecalledtheequationsofstructureofaffinespace.Therelevanceoftheequationsofstructurethusliesinthefactthattheyaretheintegrabilityconditionsoftheconnectionequations,(2.8).Theyarethebasisforthegeneralizationofaffinegeometrytodifferentiablemanifoldswithanaffineconnection,whicharetheCartaniangeneralizationofaffinespace.Inordertogetperspective,weshallnowmakeasmallincursionintothatgeneralization,whichisthesubjectofchapter8.Differentiablemanifoldsendowedwithaffineconnectionalsohaveassociatedframebundles.Assumeweweregiventheωiandωjinthebundleofonesuchkgeneralization.Ifweweretoperformthecomputationsofthelefthandsidesof(3.5),wewouldnotgetzerosingeneral.Wewouldgetlinearcombinationsoftheωl∧ωp,butnotoftheωk∧ωrandωl∧ωpterms,orelsewewouldbesmqdealingwithsomestructurestillmoregeneral.Thegeneralizationof(3.5)toouraffineconnectionswillthenread:dωi−ωj∧ωi=Riωk∧ωl,dωj−ωk∧ωj=Riωl∧ωm.(3.6)jkliikjlmOneusesthetermstorsionandaffinecurvaturetorefertotherighthandsidesofequations(3.6)respectively,andrepresentsthemwiththesymbolsΩiandΩi:jΩi≡Riωk∧ωl,Ωi≡Riωl∧ωm,(3.7)kljjlm

1095.4.CURVILINEARCOORDINATES:HOLONOMICBASES91ThisistypicalwhenoneknowstheRi’sandRi’sbutnottheωkandωr.kljlmsSometimes,onereferstothedωi−ωj∧ωiandthedωj−ωk∧ωjthemselvesasthejiiktorsionandthecurvature.Sincecomputationswiththeexplicitformthatωjandωitakeinthebundlewouldbelaboriousingeneral,oneactuallycomputesjinsectionsofthebundle,aconceptwhichweareabouttointroduce.TheΩiandΩimaybeviewedasdifferentialformcomponentsofa(1,0)−tensorandj(1,1)−tensorrespectively.Butthisisonlyjustonewayoflookingatcurvatureandtorsion.Impatientreaderscanrefertosections5.9and8.13fortheirLiealgebravaluedness.Letusreturntoaffinespace.TheequationsofstructureofaffinespacearethenwrittenasiiΩ=0,Ωj=0.(3.8)Onethensaysthataffinespacesareparticularcasesofdifferentiablemanifoldsendowedwithaffineconnectionswhosetorsionandcurvaturearezero.Aninterestingexerciseatthispointwiththeequationsofaffinespaceisthefollowing.SubstituteEqs.(2.7)in(3.5).UseAjBk=δkandreachtwotriviallyijilookingidentities,0=0.Ingeneralizedspaces,theywilltakeinterestingforms.Weshallthenusethistrivialidentitytosimplifycalculations.5.4Curvilinearcoordinates:holonomicbasesAsectionoftheaffineframebundleisacontinuousmapS:Afn→FnsuchthatP◦Sistheidentity.Insimplerterms,itisamapthatpicksoneandonlyonepointfromeachfiber.Sincethosepointsareframes,thesectionisaframefield,whichweshallassumetobenotonlycontinuousbutalsodifferentiabletowhicheverordermaybeneeded.TheconceptisequallyapplicabletothebundlesofgeneralizationsofaffineandEuclideanspaces.nPAfFigure4:SectionofFnAsweshallseeinthenextparagraph,sectionsemergenaturallywhenoneintroducesinthebasemanifold(i.e.affinespaceinthischapter)curvilinearco-ordinatesthroughaninvertibletransformation:xi=xi(A1...An),i=1...n.Needlesstosaythatwemaydefinenewcurvilinearcoordinatesintermsofother

11092CHAPTER5.AFFINEKLEINGEOMETRYcurvilinearcoordinatesratherthandirectlyfromtheAi.Thexiarenolongerrectilineariftheequationsofthetransformationarenotlinear.Inexperiencedreadersshouldbeawareoftheinconsistencyinterminologyfromoneareaofmathematicstoanother.Oneshouldusethetermaffineequationstorefertowhatareusuallycalledlinearequations.Lineartransformationyieldsthespeciallinearequationswhoseindependenttermsareallzero.Curvilinearcoordinatesystemsdefinepoint-dependentbasesofvectorseibymeansof∂Pei≡.(4.1)∂xiWeusethesametermeiasbeforesincetheseframefieldsaresectionsofthebundleofframes.Onesaysthattheeiof(4.1)arepull-backs(tosections)oftheeiof(2.3).CurvilinearcoordinatesarenotcomponentsofPorsomeothervectoruponsomebasis{bi}and,infact,onefirstdefinesthosecoordinatesandthenabasisfieldassociatedwiththem.Thebasisvectorseiandajarerelatedby:∂P∂Aj∂Ajei==aj.(4.2)∂Aj∂xi∂xiThepartialderivativesarenotconstantsbutfunctionsofthecoordinates,un-lessweweredealingwithatransformationbetweentworectilinearcoordinatesystems.Tobeprecise,wearedealingherewithcompositefunctions,sincetheAiarethemselvesfunctionsthatassignnumberstothepointsofthemanifold(Refertopoint(c)ofsection6ofchapter1).Becauseofitsimportance,werepeatthat,whendealingwithcoordinates,weusethesamesymbolforcoor-dinatefunctionsandforthevaluethatthosefunctionstake.Whetherweareconsideringtheonesortheotherswillbeobviousfromthecontext.ThenatureofthecoordinatesasfunctionsonthemanifoldstartedwiththedefinitionofPasaprojectionmaponAfn.ItfollowsthatrectilinearcoordinatesarealsofunctionsonAfn.Curvilinearcoordinatesbeingfunctionsoftherectilinearcoordinates,theyalsoare(composite)functionsonAfn.The(ei)varyfromonepointtoanother.Eachcoordinatesystemdeterminesavectorbasisateachpointoftheregionofaffinespacethatitcovers(Nobasisisdeterminedattheoriginofthesystemofpolarcoordinates,sincethissystemdoesnotassignauniquepairofnumberstoit;thedefinition(4.2)failsinanycase).Wearethusdealingherewithfieldsofbases,notjustonebasisaswasthecasewith(ai).But(ai)cannowbeviewedasaconstantbasisfield.Thisbecomesobviouswhenweobtainasystemofrectilinearcoordinatesfromanotheroneinthesamewayasforcurvilinearcoordinates.Thevectorbasisfieldsgeneratedintheprecedingwaybythesystemofcurvilinearcoordinatesisreferredtoasholonomicorcoordinatebasisfield.Inthatchangefromrectilineartorectilinear,thenewbasisfield(bi)appearsasaparticularcaseofcurvilinearbasisfieldthathappenstobethesameevery-whereintermsofthe(ai),andthusequaltoitselffromonepointtoanother.Theargumentcanbereversed,meaningthatwecouldhavestartedthewhole

1115.4.CURVILINEARCOORDINATES:HOLONOMICBASES93argumentwiththebasis(bi)andhaveobtainedthebasis(ai)asaconstantframefieldalsodenoted(ai).Non-holonomicbasisfieldswillbeintroducedinthenextsection.Weleaveforthenthepush-forward(aconceptthatamountsheretoarecovery)oftheinvariantformsinthebundlefromthedxkandωjjustconsidered.iTheuseofaparticularcoordinatesystemdoesnotforceustousecor-respondingholonomicbasisfields(unlikeinthemostcommonversionofthetensorcalculus,versionwherebasesarenotexplicitlyused).Fromthedefinition(4.1)ofei,wehave:dP=dxie.(4.3)iAlongeachcoordinateline,allthedxi’sbutonearezero.Equation(4.3)thenshowsthatthevectoristangenttothecoordinatelinexi.InasimilarwaytotheobtainingofEq.(4.2),wewouldget∂P∂xj∂xjai==ej.(4.4)∂xj∂Ai∂AiThisshowsthatthevectors(ej)Pconstituteabasissinceanyvectorcanbeexpressedasalinearcombinationoftheei’sthroughtheai’s(andthenumberofvectorsejisthesameasthedimensionofthevectorspace).Letuspracticesomeoftheseideasusingpolarcoordinatesintheplanex=ρcosφ,y=ρsinφ.(4.5)Weuse(4.2)with(x,y)and(ρ,φ)replacingtheAi’sandxi’srespectively.Wethushaveeρ=cosφa1+sinφa2eφ=−ρsinφa1+ρcosφa2(4.6)wherewehaveuseda1anda2foriandj.Noticethat,withthisdefinitionofej,dPisnotgivenbythefamiliardP=dρeρ+ρdφeφ,whichcorrespondstoorthonormalbases,butratherbydP=dρeρ+dφeφ,(4.7)inaccordancewithEq.(4.3).Theconceptoforthonormalbasesdoesnotevenenterheresinceadotproductorametrichasnotbeendefined.Thus,althoughρandφhavemetricconnotation,weignoreithereandconsiderthosefunctionsof(x,y)asanyotherfunctionswithoutsuchconnotation.From(4.6),wereadA1=cosφ,A2=sinφ,A1=−ρsinφ,A2=ρcosφfor1122thevectorbasis(onthesection)thatisabovethepoint(ρ,φ)ofAf2.Exercise.Obtaineiintermsofaj,andviceversa,forthesphericalcoordi-nates:yx2+y2r=x2+y2+z2,φ=arctan,θ=arctan.(4.8)xz

11294CHAPTER5.AFFINEKLEINGEOMETRYItisnotconvenienttoinvertthechangeofbasisbyinvertingthematrixthatgivesthischangeiftheinversecoordinatetransformationisremembered.Thus,inthiscase,wewouldproceedinthesamewaybutstartingfromthetransformationsx=rsinθcosφ,y=rsinθsinφ,z=rcosdθ.(4.9)Noticethatwedonotneedtodrawfigurestorelatedifferentvectorbases,whichmaynotbeeasytodoforunusualcoordinatesystems.Wenextsummarizemuchofwhathasbeendoneinthissection.Wear-bitrarilychoseapointofaffinespaceandintroducedanarbitrarybasis(ai).Rectilinearcoordinates(Ai)wereobtainedascomponentsofPrelativetothatbasis.Thiswasfollowedbytheintroductionofcurvilinearcoordinatesasfunc-tionsoftherectilinearones.Wethengeneratedbasisfieldsassociatedwiththecoordinatesystem.Wecompletedtheprocessbyfinallyviewingtherectilin-earcoordinatesastheparticularcasesofcurvilinearcoordinatesthatgenerateconstantvectorbasisfields{ai}.Theuseofsections—thusofaspecificvectorbasisateachpointofaffinejspace—hastheimplicationthattheωi’sarenowlinearcombinationsjjkωi=Γikdx(4.10)ofthedifferentialsofthecoordinatesinaffinespace.WethushaveidP=dxei(4.11a)jjkdei=ωiej=Γikejdx.(4.11b)Consistentlywithconceptsalreadyadvanced,theseequationspertaintoquan-tities(ontheleft)thataresaidtobepull-backstosectionsofquantitiesinthebundle.Onecanshowthatthepull-backsofexteriorderivativesandexteriorprod-uctsaretheexteriorderivativesandexteriorproductsofthepull-backsofthequantitiesbeingexteriordifferentiatedandexteriormultiplied.Weleaveforsection6thepull-backoftheequationsofstructuretosectionsofthebundle,whetherholonomicsectionslikethosewehavejustseen,oranholonomicones,abouttobeconsidered.Finally,thebasisfieldsoftwodifferentcoordinatepatchescanberelatedinawaysimilarto(4.4).Let(xi)and(xi)betwosetsofcoordinates.Weshalldenoteby(ei)thebasisfieldforthecoordinatepatch(xi).Achangeofcoordinatevectorbasisfieldisthengivenby∂P∂xj∂xjei=∂xj∂xi=∂xiej(4.12a)∂P∂xj∂xjei=∂xj∂xi=∂xiej.(4.12b)

1135.5.GENERALVECTORBASISFIELDS955.5GeneralvectorbasisfieldsAconstantbasisfield(a)togetherwithamatrix-valuedfunctionAiijofthecoordinatesAk(withdetAij=0)definesafieldofbasesbye=Aik)a.(5.1)jj(AiWedefinecorrespondingdifferentialformsωjbyusing(5.1)indP=ωje=dAia(5.2)jiandcomparingcoefficients.Invertingtheserelationsyieldstheωj’sintermsofthedAi’s.LetusdenotethesecoefficientsasAj:iωj=AjdAi.(5.3)iThisreducestothecaseintheprevioussectionifωjisdxj.Butthisispossibleifandonlyifthedifferentialsystemdxj=AjdAi.(5.4)iisintegrable.Theintegrabilityconditionsare:jj∂A∂Ai=k(5.5)∂Ak∂Aiforallvaluesoftheindices.Ifthisconditionisnotsatisfied,thebasisfield(e)iiscalledanholonomicornon-holonomic.Theprecedingpresentationofanholonomicbasisfieldscouldequallyhavetakenplacebydefiningtheωjintermsofthedifferentialsofcurvilinearcoor-dinates(xj),ratherthanrectilinearones.Insteadof(5.1),wewouldhavee=Aik)e(5.6)jj(xiAgainforthepurposeofcontrastofaffinespacewithitsCartaniangeneral-izations,letusremarkthefollowing.Inthegeneralizations,nobasisfieldcanbegivenintheform(4.1)becausethereisnoPtobedifferentiated.IfandwhenweusedPinthatmoregeneralcontext,itwillbejustnotationforsomevector-valueddifferentialform,nottheresultofdifferentiatingsomevectorfield.Asanexampleofintroductionofanon-holonomicframefieldwehavetheor-thonormalframefieldwithbases(ˆeρ,ˆeφ)associatedwiththepolarcoordinates.Inthiscase,Eqs.(5.1)nowreadˆeρ=cosφa1+sinφa2ˆeφ=−sinφa1+cosφa2.(5.7)WethenwritedPasdP=dρˆeρ+ρdφˆeφ.(5.8)

11496CHAPTER5.AFFINEKLEINGEOMETRYThevectorbasisfieldcorrespondingtoω1=dρ,ω2=ρdφisanholonomic.ThejcoordinatesAiofintersectionofthissectionwiththefiberat(ρ,φ)becomethefollowingfunctionsA1=cosφ,A1=−sinφ,A2=sinφ,A2=cosφ.Needless1212tosaythatwecouldhavechosenasystemofcoordinatesinthefiberssuchthatthecoordinatesof(ˆe,ˆe)wereA1=1,A1=0,A2=0,A2=1.ρφ1212Onceagain,thecoefficientsofthedei’sintermsofthebasisfielditselfwilljbedenotedasωi:jdei=ωiej,(5.9)ji.e.asforholonomicbasisfields.Correspondingly,wecouldexpressωiasalinearcombinationjjkωi=Γikω.(5.10a)jHowever,forthepurposeofexteriordifferentiatingωitisbettertohaveitiexpandedasjjkω=Γdx,(5.10b)iikjeveniftheωipertaintoanon-holonomicsectionofthebundle.jjBoththeΓandtheΓarefunctionsofonlythecoordinatesofthebaseikikspace,i.eofaffinespace,sincewearedealingwithasectionofthebundleratherthanonthefibersoronthewholebundle.jTheΓwillbecalledmixedcomponentsoftheconnection.Thereasondoesiknothavetodowithhavingsuperscriptsandsubscripts,butratherthat,fortheethatgoestogetherwithωjin(5.9),wehavedP=ωje,butnotdxje.Wejijjstatethisfactbysayingthatdxjandearenotdualofeachother,orthattheyjdonotcorrespondtoeachother.Beawareofthefactthatinthisparagraphtheuseofprimedquantitieswilljnotbethesameasinthepreviousone,inparticularinthecaseofΓ.Theikjrelationbetweentheωiintwodifferentsectionsisobtainedbydifferentiating(5.6).Wegetrlrlldei=Aider+dAiel=Aiωrel+dAiel.(5.11)AfterwritingdeasωjeandeasAje,weproceedtoequatecoefficientstoiijlljobtainωj=Arjωl+dAlj.(5.12)iiAlriAlBecauseofthelastterm,ωidoesnot“transform”linearly.ThesamecommentjjappliestotheΓ,since(5.12)inturnyieldsthefollowingtransformationofikconnectioncoefficientsΓj=ArjAml+AlAj,(5.13)ikiAlkΓrmi/klwhere,recallingourprevioususe“/”ofasubscriptswehavedefinedAli/kthroughdAllωk,andwherewehavefurtherusedthatωl=Γlωm=i=Ai/krrmΓlAmkandthatωj=Γjωk.rmkωiik

1155.6.STRUCTUREOFAFFINESPACEONSECTIONS97Exercise.Verifythat,ifthetwobasesareholonomic,(5.13)isequivalentto∂xl∂xj∂xq∂2xl∂xjΓj=Γm+.(5.14)ik∂xi∂xm∂xklq∂xi∂xk∂xlGivenabasisfield{e},theaffineframebundleisrepresentedby(P,A{e}P)forallPandallthenon-singularn×nmatricesA.Curlybracketsareusedforcolumnmatrices.Wecouldhavechosenanotherfieldandhavebuiltthebundleinthesameway:(P,A{e}).AlthoughthetworepresentationsusefieldsPandcoordinates,theyconstitutethesame(total)setofframes.Foranyoneofjthem,itscoordinatesAidependonthesectiontowhichweappliedthelineargroup.Theobtainingofthepush-forwardoftheωifromasectiontotheframebundleshouldbeobviousbynow:directlyapplythematricesofthefulllineargrouptotheωiwrittenasacolumn.5.6StructureofaffinespaceonSECTIONSInsection5.3,(P,e)representsn+1projectionmaps,P:Fn→Afn→Vniande:Fn→Vn.ThesemapsarefunctionsonFn.Ontheotherhand,ithevectorfieldseofsection4.4arefunctionsonAfn.Theycan,therefore,ibedenotedase(P).Oneoftenusesthenotquitecorrectnotatione(xj),oriisimplyei(x).Sincethepull-backofthederivativeisthederivativeofthepull-back,wehavedP(x)=ωi(x)e(x),(6.1)ijdei(x)=ωi(x)ej(x).(6.2)And,sincethepull-backoftheexteriorproductistheexteriorproductofthepull-backsofthefactors,wecanwritethepull-backoftheequationsofstructureinthebundle,(3.5),asijidω(x)=ω(x)∧ωj(x),(6.3)jkjdωi(x)=ωi(x)∧ωk(x).(6.4)Theseequationsthusgiveusthestructureofaffinespacefromasection’sper-spective.Theywillbewrittenwithoutexplicitindicationofthedomainsofthefunctions,i.e.asin(3.5).Again,weshallknowfromthecontextwhetherweareworkingintheframebundleoronaparticularsection.Whenadifferentiablemanifoldendowedwithaconnectionalsohasametricdefinedonit,therearesomeveryusefulnon-holonomicbasisfields.Whetheroneusestheonesortheothersdependsonwhatoneisgoingtodo.Evenifasectionisnon-holonomic,itispreferable,aswealreadysaid,touseΓjdxkikratherthanΓjωkifoneisgoingtoexteriordifferentiateωj,sinceddxk=0.ikiButoneshoulddosoonlytemporarilyandthenreturntothenon-holonomicbases.Weproceedtoexplain.

11698CHAPTER5.AFFINEKLEINGEOMETRYAnimportantdisadvantageofusingholonomicsectionsisthatdωi=ddxi=0andEq.(6.3)thenbecomesdxj∧Γidxk=(Γi−Γi)(dxj∧dxk)=0,.j

1175.7.DIFFERENTIALGEOMETRYASCALCULUS995.7Differentialgeometryascalculusofvector-valueddifferentialformsTensorcalculusandmodernversionsofdifferentialgeometryfailtoshowthatclassicaldifferentialgeometryislittleelsethancalculusofvector-valueddiffer-entialforms.Inordertoeliminatesomeclutter,thetermvector-valueddifferentialformmeansherevector-field-valueddifferentialform,sincetheyaredefinedovercurves,surfaces,etc.,notatpoints.Letavectorfieldberepresentedasv=Via=vle,(7.1)ilwhere(a)butnot(e)isaconstantbasisfield.DifferentiationofViaistrivial.iliExercise.JustifytheLeibnizruletodifferentiatevle,usingωieforde.Hint:llilinvlereplaceeintermsofa,differentiate,expresstheresultintermsofellilandcomparewiththeresultofapplyingtheLeibnizrule.BasedontheLeibnizrule,wehaveikikidv=dvei+vdek=(dv+vωk)ei.(7.2)Sincetheexpression(7.2)isvalidforanybasisfieldandalwaysequaltod(Via),iwehaveikiikiikidv=(dv+vωk)ei=(dv+vωk)ei=(dv+vωk)ei=...(7.3)Thisshowsthatthequantities(dvi+vkωi)transformlikethecomponentsofakjvectorfield.Thesocalledcovariantderivativesv,;kjjljv≡v+vΓ,(7.4);k/klkarethecomponentsofthevector-valueddifferential1−formdv,jkdv=v;kωej.(7.5)Byvirtueof(7.3),(dvi+vkωi)eisaninvariant,butdvieandvkωiearenot.kiikiThisisobviousbecausevkωieiszeroinconstantbasisfieldsbutnotinotherkiones,sinceωidoesnottransformlinearly,whichimpedesthatωialsobezerokkinotherframefields.Differentiatingagaindv,weobtainavector-valueddifferential2−form,ddv.Itscomponentsarenotwhatthetensorcalculuscallscovariantderivatives.Forjkthat,wewouldhavetodifferentiatev;kφej.Resortingto(7.2),weobtainfiveterms,twofromthedifferentiationofdvie,andthreemorefromthedifferen-itiationofvkωie.Thefirstterm,ddviisobviouslyzero.Thesecondandthirdkitermscanceleachotherout.Wethushaveddv=vkd(ωie)=vk(dωi−ωj∧ωi)e.(7.6)kikkji

118100CHAPTER5.AFFINEKLEINGEOMETRYTakingintoaccountthesecondofequations(3.6),wegetddv=viRj(ωk∧ωl)e.(7.7)ikljThecomponentsofthevector-valueddifferential2−formsddvareviRj.WeikljdefineanobjectwhosecoefficientsaretheRasfollows:ikl≡Rj(ωk∧ωl)eφi,(7.8)ikljiwhere(φ)isthefieldofbasesoflinear1−formsdualtothebasisfield(ej).isaninvariantobjectbecauseitsactiononanarbitraryvectorfieldvyieldstheinvariantddv:v=Rj(ωk∧ωl)eφivme=viRj(ωk∧ωl)e=ddv,(7.9)ikljmikljiiwherewehaveusedthatφem=δm.Although,followingcustom,wehavejusedthetermaffinecurvaturetorefertoΩ,weshouldhavereservedthisnameifor,which,beinggivenbyji≡Ωiφej,(7.10)isa(1,1)-tensor-valueddifferential2−form.Itcanalsobeviewedasa(1,1)-tensorwhosecomponentsaredifferential2−forms.Wepreferthefirstofthesetwooptions,i.e.differentialformbeingthesubstantiveandtensor-valuedbe-ingtheadjective.Noticethatthecovariantvaluednesshasbeeninducedandaccompaniedbythecontravariantvector-valuedness.Althoughthisisthefirsttimewehavementionedcovariantvector-valuedness,jithadalreadyappearedinωi.Butwerefrainedfrommentioningituntilnowjiinordertoavoidpromptinginexperiencedreadersintothinkingthatωiφejisaninvariant,whichisnot.Readerscanstartfiguringoutwhyinviewofthejlastpartofsection5.Weshallconsideragainthenatureofωiinseveraltimesinthefuture.OneobtainsthatiszeroinaffinespacebysimplydifferentiatingViaitwice.Onceagain,weshalllaterintroducestructureswhereddvisnolongerzeroandwhere,ingeneral,therulethatddiszerodoesnotapplytodifferentialformsthatarenotscalar-valued.eiandddeiarenotinvariantunderachangeofsection.Thereasonisobvious:eiforgivenvalueofirepresentsdifferentvectorsindifferentsections.ddeiwillbeconsideredinalaterchapter,inthecontextofnon-zerocurvature.Letuscomparedvwithanotherverysignificantvector-valueddifferential1−form,dP.WehavedP=ωie=δiωje.(7.11)ijiWhereasthedifferentiationofdPyieldsthetorsion,Ωie,thedifferentiationofidvleadsustotheaffinecurvature.Thismightlooksurprisingifoneweretojview(7.11)asaparticularcaseof(7.5),butwhereasthecomponentsδarekjconnection-independent,thecomponentsvareconnection-dependent.;kjkForfurtherpractice,letusdealwiththedifferentiationofv;kφej.First,kandjustincase,letusemphasizethatφejisatensorproduct,thesymbolfor

1195.8.INVARIANCEOFCONNECTIONDIFFERENTIALFORMS101suchproductbeingignoredasinmostoftheliteraturewhenwemultiplycopiesofdifferentspaces.Observealsothatonewritesωieratherthenωi⊗e,eveniithoughthatiswhatonemeans.WeapplytheLeibnizrulefordifferentiationoftensorproductsandobtain,iiusingφ(aj)=δj,iiidφ=−ωjφ.(7.12)iThismeagertreatmentofdφisanexcusetobringtoyourattentionthefollow-ingremarks.Euclideanandpseudo-Euclideangeometry,whethergeneralizedornot,isnotonlyourtrueinterestbutalsothemainsubjectofgeometricinterest.Itiscustomarilybutnotnecessarilydealtwithintheliteratureasanoutgrowthofaffinegeometry.Thisisduetohistoricalreasons.InordertotrulyaddressEuclideangeometrywithaEuclideanperspective,oneshoulduseabinitioal-gebraspecifictoEuclideanspaces,whichisCliffordalgebra.Thisalgebrahasnotyetoccupiedtheroleitshouldhaveinmathematicalteaching.Butweshallatleastbeabletoreplacetheevaluation“φi()”withtheproduct“ei·”whenweshalldoEuclideangeometryinthenextchapter,wherethereplacementfor(7.12)bestfits(Seesection3ofthatchapter,andspecificallyequation3.3).Asaconcessiontotensorcalculusbecauseofitsdominanceofthephysicsjkliterature,letusdifferentiatev;kφej.WeusetheLeibnizruleasitappliestotensorproductsandobtaind(vjφke)=dvjφke−vjdφle+vlφkde.(7.13);kj;kj;lj;kllReadersareinvitedtosubstitutepertinentexpressionsfordφanddelin(7.13)andobtainwhat,inthetensorliterature,goesbythenameofsecondcovariantjkderivatives:thecomponentsofthe(1,1)-valueddifferential1−formd(v;kφej),notofddv,whichiszerointhiscase.Inthededicatedsection12ofchapter8,weshallseewhycovariantderivativesarehorrible,exceptfor(first)covariantderivativesof(p,q)-valued0−forms,i.e.oftensorfields.Thestructuresintroducedinthischapterallowusalreadytoaddressmoreinvolvedcomputationalcases(Inordernottooverwhelmreaderswithcalcula-tions,indetrimentofstructuralconsiderations,weshallintroducethemlittlebylittleindifferentchapters).Afterall,equationstobeintroducedlateronman-ifoldsendowedwithaffineconnectionsalsoapplyhere,butsettingeverywheretozerobothtorsionandaffinecurvature.5.8InvarianceofconnectiondifferentialFORMSThetermconnectiondifferentialformshasrestrictedandcomprehensivemean-ing’s,namelytoreferto(ωj)and(ωk,ωj),respectively.Hereweshallreferiitothesecondoftheseoptions.Both(ωj)and(ωk,ωj)areinvariantsintheiibundle.But,whatdoesitmean?Westartbymakingabstractionof(i.e.ignoring)therichstructureofframebundlesandofthenatureofitselementsasframes.Whatresultsiscalledthetopologicalframebundle.Ifwerefibratethesetofbasesofourbundleover

120102CHAPTER5.AFFINEKLEINGEOMETRYthetopologicalframebundle,thefibersareconstitutedbyjustoneelement,asthegroupinthenewfibersonlycontainstheidentitytransformation.Theconnectiondifferentialformsarethensaidtobescalarsorinvariantsbecausetheyobviouslydonotchangeunderthetransformationsinthefiber,whichonlycomprisetheidentitytransformation.Readerswhofindthissecondbundletootrivialtobehelpfulneedconsiderathirdfibration,intermediatebetweenthestandardfibrationandtheonewehavejustintroduced.ItistheFinslerianfibration.However,giventhatlin-eartransformationshavelessintuitivegeometricinterpretationsthanrotations,wearegoingtoprovidetheFinslerianrefibrationofbundlesoforthonormalframes.Orthonormalframes(i.e.madewithorthonormalbases)belongtothenextchapterandsomereadersmayconsiderpostponingfurtherreadingofthissectionuntillateron.Formanyotherreaders,knowledgeofspecialrelatively(SR)willallowthemtodealatthispointwithabundlethatwehavenotyetintroduced.Considerthesetofalltheinertialbasesatallpointsofrelativisticflatspace-time.Thebasesateachpointarerelatedamongthemselvesbytheactionofthe(homogeneous!)Lorentzgroupinfourdimensions.Thesefibersconstitutedifferentiablemanifoldsofdimensionsix,allofthemidentical.Thecoordinatesonthefiberscanbetakentobethreevelocitycomponentsviandthreerotationiparametersφ.Thesesixcoordinatestogetherwiththefourspacetimecoordi-nateslabelalltheframesinourset.Therequirementwemadethat,fromaphysicalperspective,thespatial(sub)baseshavetobeinertialisnotessential;ithasbeenmadeinordernottogetdistractedwithotherissues.Theaforementionedrotationsconstituteathreedimensionalsubgroupofthe10-dimensionalPoincar´egroup.ThequotientofthePoincar´egroupbythatrotationgroupisasevendimensionalsetlabelledbyspacetimeandvelocitycoordinates.Itcanbetakenasabasemanifoldfora“Finslerian”refibrationofthe10-dimensionalsetofframes,withthethree-dimensionalrotationgroupactingonthefibers.Wenowneedtoconsiderhowtheconceptoftensorialityappliestosectionsofthisbundle.Wesaythatasetofquantitiestransformsvectoriallyunderagroupoftrans-formationsofthebasesofavectorspaceif,underanelementofthegroup,theytransformlikethecomponentsofvectors,whethercontravariantorcovariant.Wesaythatsomequantityisascalarifitisaninvariantunderthetransfor-mationsofagroup.Aquantity(respectively,asetofquantities)maybescalar(respectivelyvectorial)underthetransformationofagroup,whilenotbeingsounderthetransformationsofanothergroup.Thusthesumviviisascalariunderthetransformationsoftherotationgroup,butnotofthelineargroup.Theinvariantsumgvivj(seenextchapter)istheaffineextensionbythei,jijlineargroupoftheexpressionvivi,whichisleftinvariantbythe(largest)irotationsubgroupofthelineargroupinquestion.Beforewereturntotheissueofrefibrations,weadvanceonemorecon-ceptaboutthesetofframesofSR.Letμbe(0,i)withi=1,2,3.Unlikeinfour-dimensionalaffinespace,therearenowonlysixindependentωνs,namelyμ

1215.9.THELIEALGEBRAOFTHEAFFINEGROUP103(ωi,ωm)withm>l,whichisthesamenumberasthedimensionalityofthe0lfibers.Thereasonisthat,asweshallseeinthenextchapter,ω0=ωiandi0ωl=−ωm.mlIntheFinslerianrefibrationoftheframesofSR,theωm(asopposedtothelωμ;λ=0,l;μ=0,m)playtheroleoftheconnection,andtheωiliveinλ0thenewbasemanifold,i.e.theonetowhichwehaveassignedthecoordinates(xμ,vi).Theωiareliketheωiinthattheydonotdependonthedifferentials0idφofthecoordinatesonthefiber,andthattheytransformlikethecomponentsofavectorundertherotationgroupinthreedimensions.ω0isinvariant,alsocalledscalar,underthesamegroup.Wereturntothestatementofinvarianceoftheωμ’sandων’swhenthesetμofframesisviewedasabundleoveritself.Allthedifferentialformsωμandωνarescalarsbecausethegroupinthefibersonlyhastheunitelement.Weμmaylegitimatelyask:istherefibrationofthebundle“overitself”justatricktocreatesuchinvarianceoutofthinair?No,itisnot.Weareabouttoseeinthenextsectionandinthenextchapterthatthe(ωμ,ωλ)arethecomponentsνofadifferential1−formtakingvaluesintheLiealgebraoftheaffineortheEuclidean(orPoincar´e)group,asthecasemaybe.TheLiealgebraisofthesamedimensionasthegroupGinthepair(G,G0)thatconstitutestheKleingeometriestowhichthoseexamplesrefer.Weneed,therefore,notviewtheconnectionassomeentitythatfailstobelongtoatensorstructure,butasanentitythatbelongstosomeotherstructure,aLiealgebraThetensorialapproachtogeometry—whichhasverylittletodowiththegroupsGandonlymarginallymorewiththegroupG0instandardpresentationsforphysicistsofthetensorcalculus—isahistoricalaccidentwhichshouldberecognizedassuch.Oncethesetofallframesofageometryisviewedasthearenawheretheactiontakesplaceindifferentialgeometry,wechangecoordinatesinit,sayfromyAtozA.TheindexAgoesfromoneton+n2inaffinespace,andfromoneton+[n(n−1)/2]inspacesendowedwithadotproduct.Take,forexample,ω2.Wehave1ω2=a2dyA=b2dzA=...,11A1Awhereyandzarecoordinatesystemsinthebundleandwherewesumoverallitscoordinates.Thatistheformthatthestatementofinvarianceofω2takes.15.9TheLiealgebraoftheaffinegroupTheLiealgebraoftheaffinegroupisnobigdeal.Itisimplicitintheconnectionjequation{dei}=[ωi]{ej}andwhatwedowithit.TheideaissimplytounderstandwhatismeantbyaffineLiealgebravaluednessoftheconnectionjdifferentialformωi.Forthisauthor,theusefulnessofdealingherewiththisconcepthastodowithcorrectingthewrongimpressionthatwhereasYang-MillstheoryhastodowithbundlesandLiealgebras,traditionaldifferentialgeometrydoesnot.Itdoes.Anaffinegroupcomprisesacorrespondingsubgroupoftranslations.They

122104CHAPTER5.AFFINEKLEINGEOMETRYarenotlineartransformationsonvectorssinceTw(u+v)=Tw(u)+Tw(v),(9.1)whereTwdenotesatranslationbythevectorw.Theycannot,therefore,berepresentedbyn×nmatrices.Theycan,however,berepresentedby(n+1)×(n+1)matrices,namely1Ai,(9.2)0IwhereIistheunitn×nmatrix,andwheretheAiarethecomponentsofthetranslationrelativetoabasisai(OurLatinindicestakeagainthevalues1ton;theymomentarilytookthevalues1ton−1intheprevioussection).Moreconcretely,wehaveP1AjQ=(9.3)ei0Iajforatranslationand1AjPQQ=g=j(9.4)eiaj0Aiajforageneralaffinetransformationg,withQandajrepresentingafixedpointandafixedbasis.Equations(2.6)arenowwrittenasdPP0ωjP−1=dg·g=j(9.5)deiej0ωiejwithωjandωjgivenby(2.7).iALiegroupisagroupthatisatthesametimeadifferentiablemanifold.TheLiealgebraofaLiegroupisnothingbutdg·g−1;ifgisthegeneralelementofanaffine,linear,Euclidean,rotationgroupforagivendimension,dg·g−1istherespectiveaffine,linear,Euclidean,rotationLiealgebra.Theyarematriceswithentriesthataredifferential1−formsInfactLiealgebrasaresometimesdefinedasmatrixalgebrasthatsoandso.Onethussaysthattheconnection(ωi,ωk)isadifferential1−formvaluedintheLiealgebraoftheaffinegroup.jTheLiealgebraofthelineargroupisreadfromj{dei}=[ωi]{ej},(9.6)thusvaluedintheLiealgebraofthelinearsubgroupoftheaffinegroup.Noticethatthesquarematrixin(9.6)resultsfromremovingthefirstcol-jumnandthefirstrowofthesquarematrixin(9.5).NevermindthatωiisalinearcombinationofthedxlandthedAm.Inonecasewearedealingwithlthemanifold(thebundle)wherethedifferentialformsaredefinedandanotheroneiswheretheytaketheirvalues(theLiealgebra).If(9.6)isobtainedbyjconsideringthelineargroupfordimensionnonitsown,theωidependonlyonthecoordinatesAmandtheirdifferentials.Butif(9.6)isextractedfrom(9.5)l

1235.10.THEMAURER-CARTANEQUATIONS105jbyeliminatingthefirstrowandcolumn,theωidependalsoonthedifferentialsdAi.Asusual,wedifferentiate(9.5)andobtainΩie0dωjP0ωk0dωjPij=j−k∧j,(9.7)Ωiej0dωiej0ωi0dωkeji.e.torsionandcurvature.Noticethatwedonotgetzerointhefirstrowofthecolumnmatrixontheright.Theexteriorproductofmatricessimplymeanstheskew-symmetrizationofthematrixproduct,whichpermitsustowriteΩie0dωj−ωk∧ωjPi=k,(9.8)Ωje0dωj−ωk∧ωjeijiikjwhereweagainseethefirstrowifwearetorepresentΩieandΩjjointlyiniimatrixform.Thishelpsemphasizethatthefirstrowalsoplaysaroleandthat,therefore,ΩitakesvaluesintheLiealgebraoftheaffinegroupandnotjustthelineargroup.ThisisdueinanycasetothefactthatΩieinvolvesthefirstrowiofthesquarematrixWereturntothemainargument.Withsummationoverrepeatedindices,thesquarematrixin(9.5)canbewrittenasjjjωαj+ωiαi,i,j=1,...,n.(9.9)i,jjwheretheαjandαiareeasilyidentifiablefrom(9.5)andconstituteabasisinthealgebraof(n+1)×(n+1)matriceswhosefirstcolumnismadeofzeroes.Eachαjhasallitselementszeroexceptfortheunitinthefirstrowandj+1jcolumn(j>0),andeachαihasallitselementszeroexceptfortheunityinrowi+1andcolumnj+1(i>0,j>0).Thespacespannedby(α,αi)isjjcalledtheLiealgebraoftheaffinegroup.Theissuearisesoftheproductthatmakessaidspaceanalgebra.Theskew-symetrizedproductofmatricesdoesit,andalsomakestorsionandaffinecurvaturebelongtotheLiealgebra.5.10TheMaurer-CartanequationsInhis1937book“Thetheoryoffinitecontinuousgroupsanddifferentialge-ometrytreatedbythemethodofthemovingframes”,CartanshowsthattheequationsoncontinuousgroupsknownasMaurerequationsrevealthestructureofthedifferentialformsdgg−1towhichthoseequationsrefer[22].Thatisthereasonwhy,hesaid,“theyareknownasequationsofstructureofE.Cartan”.´Nowadays,theyarecalledMaurer-Cartanequations.TheMaurer-CartanequationsstatethatdωA=CA(ωB∧ωC),(10.1)BC

124106CHAPTER5.AFFINEKLEINGEOMETRYwheretheωDrepresentabasisfortheLiealgebradgg−1.Thepurposeoftheparenthesisin(10.1),whichisrelatedtowhatvaluesoftheindicesthesumrefersto,isthesameasinpreviousoccasions.Therelevanceoftheseequations,whichwouldotherwisebetrivial,isthattheCAareconstants,specificallyBCcalledstructureconstants.TheuseofcapitalsfortheindicesservestocallattentiontothefactthattheωArefernotonlytowhatwehavereferredastheωμ(orωi),butalsototheων.ReaderscanreadtheCAfortheaffinegroupμBCanditslinearsubgroupfromthealreadygivenequationsofstructure.TheMaurerequationsarefrom1888,butnotexactlyaswehavewrittenthem,sincetheexteriorcalculuswasnotbornuntil1899.Oneonlyhaddiffer-ential1−forms,calledPfaffian,andsomethingcalledtheirbilinearcovariants,whicharerelatedtotheirexteriorderivatives.Wementionitsothatread-erswillseethesocalledpropertiesoftheconstantsofstructurefromthenewperspectivethattheexteriorcalculusbringstothem.Sincewearegoingtodifferentiate(10.1),itisusefultorewriteitasA1ABCdω=CBCω∧ω.(10.2)2TheCAdonotdeterminetheCA,these(thenon-nullonestobeprecise)BCBCbeingdoubleinnumberifweaddtheconditionCA=−CA.(10.3)BCCBHalfofthemareequaltotheCA,andtheotherhalfaretheiropposite.WeBCcannowdroptheprimes.Differentiationof(10.2)yieldszeroontheleft,andalsoforthetermwherewedifferentiateCA.Thelasttwotermsareequal.Substitutionof(10.2)inBCthosetermsyields:0=CACBωD∧ωE∧ωC,(10.4)BCDEand,therefore,0=CACB+CACB+CACB−CACB−CACB−CACB.(10.5)BCDEBDEcBECDBCEDBDCEBEDCThefirst,secondandthirdtermsarerespectivelyequaltothefourth,fifthandsixthterms.Equations(10.5)canthusbewrittenas0=CACB+CACB+CACB.(10.6)BCDEBDECBECDWhereasCA=−CAisthedefinitionofthesymbolsononesideintermsBCCBofthesymbolsoftheotherside,(10.6)isanactualproperty,which,aspresented,hidesitsinnersimplicity,namelyddωA=0(10.7)fortheinvariantformsofthegroup.Weleaveitforthereadertofigureoutthetrivialgeometricidentitiesthat(10.7)correspondto.

1255.11.HORIZONTALDIFFERENTIALFORMS107LetusnowseewhatdifferenceCartan’sgeneralizationofaffinespacemakes.Inthebundle,theωlareindependentoftheωj.HencetheRjωk∧ωlarenotiikloftheformCjkmωl∧ωp,andtheωjarenot,therefore,theinvariantformsilpkmiofthelinearsubgroup.Thisremarkhastobeseeninthecontextthat,aswejshallseeinchapter8,onlythepull-backstofibersoftheωi’sofdifferentiablemanifoldsendowedwithanaffineconnectionareinvariantformsofthelinearjgroup,nottheωithemselves.Thecaseofthefirstequationofstructureisalittlebittrickybecausethefirstequationofstructureismoreremotelyconnectedtothetranslationsthanthesecondequationofstructureisrelatedtothelineartransformations.Inthefirstplace,theMaurer-Cartanequationofthetranslationgroupcannotcontainjtheωi(Noticethat,incontrasttothefirstequationofstructure,thesecondjjequationofstructureofaffinespacecontainsonlythetheωi,notboththeωiandtheωl).SotheissueofgeneralizationofthefirstequationofstructureofaffinespaceisnotanissueofgeneralizationoftheMaurer-Cartanequationofthetranslationgroup.5.11HORIZONTALDIFFERENTIALFORMSThesophisticatedconceptofhorizontalityconcernsthebundleasawhole,notitssections.Afterpullingdifferentialformsfromabundletoitssections,theconceptcannolongerbeformulated.Saidbetter,thepull-backtoasectionofahorizontalformofthebundleisatensorialobject.Theconceptofhorizontalityisusefulforunderstandingthatsomethingthatlookslikeatensormaynotbeone.Inthebundle,wecanexpressadifferential1−forminthefollowingalterna-tiveways:α=P(x,A)dxm+Pk(x,A)dAi=Q(x,A)ωr+Qk(x,A)ωi.(11.1)mikrikThelastofthesetwoexpansionsofαhasanintrinsicsignificancethatthefirstonelacks,significancethatisduetotheinvariantcharacterof(ωr,ωi)inthekframebundle.ThatisparadoxicallyduetothefactthattheωiisalinearkcombinationofthedxmanddAi,buttheωrisnotafunctionofthedAi.ThekktermP(x,A)dxmhascontributionsnotonlyfromωrbutalsofromωi,themklattercontributionnotbeingcompensatedbyacontributiontoPk(x,A)dAiikfromQ(x,A)ωr.rOnereferstoQ(x,A)ωrasthehorizontalpartofα.Laterinthissection,rweshalldiscusscovariantderivativesfromthisperspective.Whowouldimaginethatinthedifferentialsofthecomponentsofavectorfieldhide(pull-backsof)termsofbothtypes,linearinωrandinωi?Theydo!kThesetofframesoftheframebundlecanbeexpressedasthepush-forwardfromanygivensection{e}.Symbolically,wehave{E}=g0{e},(11.2)

126108CHAPTER5.AFFINEKLEINGEOMETRYwhereg0isthelineargroupandwhere{e}issomebasisfield.Ofcourse,we−1shouldhavereplacedg0withG0,but,aswhenwewrotedg0g0,wearefollowingnotationalcustomindealingwiththistopic.Differentiating(11.2),wehave:−1d{E}=dg0{e}+g0d{e}=dg0g0{E}+g0ω{e}−1−1=(dg0g0+g0ωg0){E},(11.3)jwhereωrepresentstheωi.Theexpressionvmeforavectorfield,v,pertainstosections(e).But,mmlikethebasisfield(em),itcanbepushedforwardtothebundle,meaningthatwecanwritevitasv=vie=VjE.(11.4)ijTheVjwillbeafunctionofthecoordinatesinthefibersbyvirtueofitsrelationtovi.Itwillalsodependonxbecausesodoesviingeneral.Weuse(11.4)todefinedifferentialformsDvmandDVm:dv=(dvm+viωm)e≡(Dvm)e(11.5a)immanddv=(dVm+Viωm)E≡(DVm)E,(11.5b)immClearly,(Dvm)isapull-backof(DVm).Equivalently,(DVm)isthepush-forwardof(Dvm).Similarlytheωmin(11.5a)isthepull-backtothesameisectionoftheωmin(11.5b),whichpertainstothebundle.Inparallelto(11.4),iwehavedv=(Dvm)e=(DVm)E.(11.6)mmTherelationbetween(Dvm)and(DVm)isimpliedbytherelationbetween(e)and(E),andisthesameastherelationbetween(vm)and(Vm),asmmcomparisonof(11.4)and(11.6)shows.Equations(11.5)-(11.6)thusimplyDVj=AjDvi.(11.7)iInadditiontodependingonAj,DVjdependsonxanddx,sincedvidoes,butinotonωj.However,dvibelongstoasection,wherethepull-backofωjentersiiDvi.Wemustthusdealwiththisissueintheframebundle,whereωjandωjiareindependent.Werewrite(11.6)asdVm=DVm+(−Vi)ωm.(11.8)iInthisequationeachandeveryquantitybelongstothebundle.ItfollowsthatDVmisthehorizontalpartofdVm,becauseofthedefinitionofhorizontalitytogetherwiththefactthatthisdecompositionisuniqueinthebundle.Abusingthelanguage,wemaysaythatthecovariantderivativesarethecoefficientsofthehorizontalpartofdvm;weshouldbereferringtothehorizontalpartofdVm,notdvm.Readerswhoneedtogointogreatergeneralitybecauseoftheirparallelinterestinthetensorcalculuswouldhavetorepeattheforegoingprocesswithtensorfieldsratherthanvectorfields.Inthisbookwearenotinterestedinthetensorcalculus.

127Chapter6EUCLIDEANKLEINGEOMETRY6.1EuclideanspaceanditsframebundleBythenameofEuclideanpointspaceorsimplyEuclideanspace,En,werefertotheaffinespacewhoseassociatedvectorspaceisEuclidean.Aswebecomeincreasinglyconscientiousoftheroleofgroupsindefiningageometry,weshallstarttoavoidstatementslike“aEuclideanconnectionisanaffineconnectionthat...”.Thepairofgroups(G,G0)involvedindefininganythingaffineisdifferentfromthepairofgroups(G,G0)directlyinvolvedindefininganythingEuclidean.ThedotproductallowsustodefinetheEuclideanbases,i.e.thosesatisfyingtheorthonormalityconditionai·aj=δij.(1.1)TheEuclideanframebundleisthebundleofallEuclideanframes,i.e.pairsofapointandavectorbasis.ThegroupGforthisbundleistheEuclideangroup,andthegroupG0initsfibersistherotationgroup,bothforthesamedimensionn.Thewholebundlecanbeobtainedfromanysectionbytheactionoftherotationgroupforthegivennumberofdimensions.Thus,forinstance,wehavethesectionep(p0)=icosφ+jsinφ,eφ=−isinφ+jcosφ(1.2)inE2.Thecircumflexisusedtodenoteorthonormalbases.Bytheactionoftherotationgroupintwodimensions,weobtain(orrecover,ifyouwill)thefullbundlefromasection.Donotconfusethepointdependentrotationofangleφdefiningthesection(SeeEq.(1.13))withthesetofallrotationsintheplanetoobtainthebundle(SeeEqs.(1.11)and(1.12)).Returningtogeneraldimension,n,letαdenotethecoordinatesinthefibers(i.e.theparametersoftherotationgroupinndimensions)andletxdenote109

128110CHAPTER6.EUCLIDEANKLEINGEOMETRYthe(ingeneral)curvilinearcoordinatesinasection.Bytheconstructionofthepreviousparagraph,weshallhaveei(x,α)·ej(x,α)=δij,(1.3)inEuclideanframebundlesandei(α)·ej(α)=δij(1.4)intheirfibers(meaningthatxisnolongerafunctionbutnumbers,absorbedinthenotation).AnaturalwaytocreatethebundleofframesofaEuclideanspaceistofirstintroduceabasisfieldby,forexample,usingvectorstangenttothecoordi-natelines,thenorthonormalizingthemandfinallyapplyingthecorrespondinggroupofrotationsateachpointofthespaceinquestion.ItmustbeclearthatthebasesineachfiberarerelatedbythegroupG0,i.e.byrotations.Inthesectionsandinthebundle,ontheotherhand,therelationbetweentheframesisgivenbythegroupG,i.e.bygeneraldisplacements,whichalsoincludetranslations.InEuclideanspace,theruletocomparevectorsatadistanceisimplicitinthefactthatallvectorbasescanbereferredtojustonevectorspaceandthusjustonevectorbasis,(i,j,k,...).Then,retrospectively,thisisequivalenttohavingsectionsofthebundleofframeswherethebasisatonepointisequaltothebasisatanyotherpoint.Ofcourse,mostsectionsarenotconstant.Thesection(1.2)isnotconstantinEuclideanspace;see,however,section2ofchapter7foradifferentiablemanifoldwhereitis.Whetherwedifferentiateequation(1.3)or(1.4),weget,usingde=ωle,iilωlδ+ωlδ=ω+ω=0,(1.5)iljjilijjiwithωdefinedasωkδ.Clearly,ijikjjiωi=−ωj,(1.6)whichimpliesthattherearenowonlyn(n−1)/2independentdifferentialformsωj.Thereweren2independentsuchformsintheaffinecase.iWhenthepositionvectorPisgivenintermsoforthonormalbasis,P=Aia,(1.7)ithecoordinatesAiarecalledCartesian(notallrectilinearcoordinatesareCarte-sian).Obviously,wehavedP=dAiaiintheseconstantsections,anddP=dAia=ωi(α,dAi)e(1.8)ii

1296.1.EUCLIDEANSPACEANDITSFRAMEBUNDLE111inarbitraryorthonormalbasesorbasisfields(ei)ofthebundle.Theαaretheparametersoftherotationgroupforthegivendimensionn.Theωi’sandtheeiaresaidtobedualtoeachother.Hadweusedanarbitrarysection(orthonormal!)tospanthebundle,dPwouldthentakethemoregeneralformdP=ωi(x,α,dx)e.(1.9)iSee,forexample,Eq.(1.12),whereαisacoordinateinthefibers.Onsections,αisasetoffunctionsα(x)and,therefore,dP=ωi(x,α(x),dx)e=ωi(x,dx)e.(1.10)iiWeproceedtoillustrateinthebundleofE2thattheωiareinvariantforms,apointmadeinmoregeneraltermsinthelastsectionofthepreviouschapter.IntermsoftheCartesianbasis(dx,dy)ofdifferentialforms,theωiofthebundlecanbegivenasω1cosαsinαdx2=,(1.11)ω−sinαcosαdyforall0≤α<2π.Intermsofthebasis(dρ,ρdφ),wesimilarlyhaveω1cosαsinαdρ2=(1.12)ω−sinαcosαρdφforall0≤α<2π.Theforms(1.11)forthesetofallαarethesameasthesetofforms(1.12)forallα.Thisisclearsincedρcosφsinφdx=,(1.13)ρdφ−sinφcosφdywhichsubstitutedin(1.12)yields(1.11)withα=α+φ.Noticetheappearanceofthecoordinateρin(1.12).Nosimilarappearanceofxorytakesplacein(1.11).Ingeneral,oneshouldnotassumethatanequationorstatementvalidintermsofCartesiancoordinatesisalsovalidwhenothercoordinatesorbasisfieldsareinvolved.Itisclearnowthat(P−P)2=AiA=Σ(Ai)2,(1.14)21iwheretheAiareinE3the(Cartesian)coordinatesx,y,z.Needlesstosay,wehaveA=Ai,whichwehaveusedin(1.14).iLetusdefineds2,calledthemetric,asds2≡dP·dP=ωiωje·e=ωiωjδ=(ωi)2=ωiω(1.15)ijijiiwithω=ωiTheseequationsbelongtoboththebundleanditssections.Itiiswellknownfromelementarycalculushowtoobtainthedistanceonacurvegiventhemetric.

130112CHAPTER6.EUCLIDEANKLEINGEOMETRYItisworthbeingawareofthefactthattheproductin(ωi)2andinωiωisaitensorproduct.Alargepartofthecontentoftheconceptoftensorproductislinearityineachofthefactors(Therearealsootherlinearproducts,likeexteriorandClifford,whicharenottensorproducts).Letusconsiderthesimpleexampleofdimensiontwo.Abasisforthemoduleoftensorsofrank2inthetensoralgebraofdifferentialformsfor2-dimensionalspace(whether,affine,Euclideanorotherwise)isgivenbyω1ω1,ω1ω2,ω2ω1,ω2ω2.Incontrast,andaswealreadyknow,itwouldbejustω1ω2(writtenω1∧ω2)forexteriorformsofgradetwo.Weshallnotdelvefurtherintothissincethereisamuchbetterwayofdoingmetricsindifferentialgeometrythanusingtensors.Justforreference:inFinslerbundles,thedistancebecomesadifferential1−form[73].Apseudo-Euclideanspaceisonewheretheassociatedvectorspaceispseudo-Euclidean.Thatmeansthatei·ei(nosummation)is−1foratleastonevalueoftheindex.Themetricthentakestheformds2=(ωi)2=ωiω,(1.16)iiiwherei=±1.Thetermsignatureisusedtorefertothenumberofindicesforwhichi=+1minusthenumberofindicesforwhichi=−1.Withdis-playofactualsignsofthei,signaturesofthetypes(1,−1,−1,...,−1)and(−1,1,...,1)arecalledLorentzian.Fordimensionn=4,thespaceiscalledtheLorentz-Einstein-Minkowskispace.Theequationstofollowwillremainthesame,independentlyofsignature,exceptwhenwewarntothecontrary.WhendealingspecificallywiththeLorentziansignature,weshalluseGreekindices,theindex“zero”beingreservedforthesignintheminorityinthesignature.For(1,−1,−1,−1),wehaveω=ω0,ω=−ωi,(1.17)0iωi=−ω=ω=ω0,(1.18)00ii0iasfollowsfromdefiningωasωμ(nosum).TheGandGgroupsnowareμμ0thePoincar´eandLorentzgroups,respectively.6.2ExtensionofEuclideanbundletoaffinebundleBytheactionofthelineargrouponthefibersofaEuclideanframebundle,we“recover”theaffineframebundleforthesamenumberofdimensions.ItiscalledtheaffineextensionoftheEuclideanframebundle.Arbitrarybasesofthisextensionsatisfyarelationoftheformei(x,α)·ei(x,α)=gij(x),(2.1)

1316.2.EXTENSIONOFEUCLIDEANBUNDLETOAFFINEBUNDLE113ratherthan(1.3).Onsections,ei(x)·ej(x)=gij(x).(2.2)In1922,Cartanexploitedequations(2.1)-(2.2)tocreategeometryinbundles.Theideaisthatthedependenceontheparametersαofagrouponthelefthandsideof(2.1)isnotpresentonitsrighthandside.Foragivenpointofcoordinatesx,abasisei(x)atthatpointgeneratesa“fiberofbases”yieldingthesamecomponentsgij(x)ofthemetric.TheextensionofthemetrictotheaffineframebundleisgivenbydP·dP=g(x)dxidxj,(2.3)ijandalternativelyasdP·dP=g(x)ωiωj,(2.4)ijwheretheωiarenotholonomic.Non-holonomicbasesarerarelyifeverusedexceptforg(x)=δ,sinceitispossibletoalwayswritethemetricas(1.16)ijijthroughdiagonalization.Inmostcasesofpracticalinterest,thisdiagonalizationcanbeachievedbyinspection.Ifthemetricisdiagonal,i.e.ifitisoftheform2iids=giidxdx,(2.5)iweobviouslyhavesectionswherei√iω=igiidx(nosum).(2.6)Ifthemetricisnotdiagonal,itcanbediagonalizedinmorethanoneway,aswealreadysaw.Differentiationof(2.2)yieldsllωiglj+ωjgil=dgij,(2.7)and,therefore,dgij−ωij−ωji=0,(2.8)withω≡ωlg.(2.9)ijiljjNoticethattheωi’sarenolongerindependentsincetheymustsatisfy(2.3).jThereareonlyn(n−1)/2independentωi’sintheaffineextensionoftheEu-clideanframebundle.jGiventhat,onsections,theωi’sareoftheformjjlωi=Γildx,(2.10)andthatdgisgdxm,Eq.(2.8)canbewrittenasijij,mllgij,m−Γimglj−Γjmgli=0.(2.11)

132114CHAPTER6.EUCLIDEANKLEINGEOMETRY6.3MeaningsofcovarianceThissectionmighthavegoneinthepreviouschapter,exceptthatEuclideanspacesprovideustheconceptofreciprocalvectorbasisfields,whichaffinege-ometrydoesnot.Thesefieldswillenterourdiscussionatsomepointinthissection.Thetermcovarianthasavarietyofmeanings.Onedoesspeakofcontravari-antandcovariantvectors(andvectorfields).Thecontravariantvectorsaretheonesofwhichwehavespokeninthepreviousandpresentchapters,exceptiwhenwebroughtuptheφ,preciselythecovariantvectorsorlinearfunctionsofcontravariantvectors.Thisterminologyisnotveryfortunatesince“contra”meansagainst.But,againstwhat?Withoutenteringdetails,letussaythatthisterminologyhasitsrootsintheearlytimesofthetensorcalculus,whenafactorofgreatrelevancewashowasetofcomponentstransformedunderachangeofcoordinates.Wehaveinmindspecificallyequationssuchas(5.1)-(5.2)ofchapter3.Inthefollowing,whenwesaytransformread“changeofbasis”.Inaffinespace,thingsaresimple.Thecomponentsofcontravariant(co-variant)vectorsaresaidtotransformcontravariantly(respectivelycovariantly).Theelementsofthebasesofcontravariant(respectivelycovariant)vectorsthentransform(watchit!)covariantly(respectivelycontravariantly).Inthatway,contravariant(respectivelycovariant)componentsarecomponentswithrespecttocontravariant(respectivelycovariant)basesofcontravariant(respectivelyco-variant)vectors.Butwhytheredundancy?Isthereaneedtospecifycontravari-ant(covariant)basesofcontravariant(vectors)?Notinaffinespace,wherethebasesaregeneralvectorbases.Butwearepreparingthewayforthenextparagraph.Wehavesaidthatcontravariantbasestransformcovariantly,i.e.oppositelytothecontravariantcomponentssothatthecontractionofthecomponentswiththebasiswillyieldaninvariant.But,inEuclideanspaces,atangentvectorcanbereferredalsotodualbases,whichtransformcontravariantly,notcovariantly.Thecomponentswiththosebasesthentransformcovariantly.Inotherwords,vectorsofEuclideanspaceshaveboth,contravariantandcovariantcomponents.Hencethebehaviorofthecomponentsofavector(orofavectorfield)nolongerspeaksofthenatureoftheobjecttowhichthecomponentsbelong,sinceitdependsonthetypeofbasisthatoneischanging.Forthisreason,itisbesttousethetermstangentandcotangentvectorsforwhatoneoftencallscontravariantandcovariantvectors.Tangentvectorcanthushavecontravariantandcovariantcomponents.Similarconsiderationscanbemadeforcotangentvectors,butweshallnotgetintothatsincewedonotfeelaneedforit,speciallyinviewofwhatwesaidaboutreplacingbasesφiwithreciprocalbasesei.Theideaofusingeiistounifytheconceptualspaces.Introducingalsoφiwouldbelegitimatebutwouldnothelpwithconceptualunification.Nextconsiderfunctionsofcurves,i.e.whatwehavereferredtoasdifferential1−forms.Theircomponentstransformlikecomponentsofcotangentvectorfields.Infact,theyaredefinedascotangentvectorfieldsinmanypublications,

1336.3.MEANINGSOFCOVARIANCE115butnothere.Oursisadifferentconcept:adifferential1−formisafunctionofcurves,notofvectors.ThisisalsothecaseinRudin[65],andalso,thoughlessexplicitlyso,inCartan’sandK¨ahler’spublications.Thereisasecondmeaningofcovariant.Oneoftensaysthatanobjectiscovariantwhenonewantstosaythatitscomponentsbehavetensorially.Thismayrefertocontravariant,strictlycovariant,ormixedcontravariant-covariantbehavior.Forexample,thecovariantderivativeofavectorfieldcombinescon-travarianttransformationpropertiesrelativetooneofitstwoindices,andco-varianttransformationsfortheotherone.Nowcomesthemostimportantremarkofthissection.Unlikethevi,which;jinvolvestheconnection,thevidonottransformtensorially,unlesstheviare,jthecomponentsofavectorfieldwithrespecttoaconstantfieldofbases.Theexteriorderivativeofadifferential1−form,ontheotherhand,doesnotdependonconnectionbutstillyieldsanobjectthathastheappropriate“tensorialorcovariant”transformationproperties.Henceitisunfortunatetospeakofthecovariantderivative,asiftheexteriorderivativedidnotgiverisetocovariantquantities.Thesecondmostimportantremarkisasfollows.AsusedbyCartan,K¨ahlerandFlanders[41],disanoperatorwhich,actingonrespectivelyscalar-valueddifferentialforms,vector-valueddifferential0−formsandvector-valueddiffer-entialr−forms(r>0)yieldswhatarecommonlycalledtheexterior,covariantandexterior-covariantderivatives.Onemightbetemptedtosaythattheeffectsoftheoperatordaredifferentdependingonwhatobjecttheyareactingon.Notso,sincescalar-valueddifferentialformsandvector-valueddifferential0−formsareparticularcasesofvector-valueddifferentialr−forms.Onceitisrecognizedthatoneonlyneedstheencompassingconceptofexteriorcovariantderivativeinordertodealwiththosederivative“s”,onemayuseinsteadsimplythetermexteriorderivative,whichiswhatCartanandK¨ahlerdo.Thedisplayofbases(asin,say,veiorvωiorvφi),helpsavoidconfusion.iiiWeproceedtodiscussdifferentiationintermsofreciprocalbasesei.Wedefinethebasis(ei)bytherelationei·e=δi,(3.1)jjregardlessofwhetherthebasis(ej)isorthonormalornot.Weobtaindei·e=−ei·ωke=−δiωk=−ωi,(3.2)jjkkjjwhichinturnimpliesdei=−ωiej.(3.3)jEquation(3.3)permitsustodifferentiateavectorfieldvwhenwrittenasvei.iIndeed,jikjkjidv=vi/jωe−vkωje=(vi/j−vkΓij)ωe.(3.4)Letuscompare(3.4)withtheexteriorderivativeofα=vωi.Fortheipurposeofdifferentiation,itisbesttousecoordinatebasesof1−forms.Itisthenclearthatdα=d(vdxi)=(v−v)(dxj∧dxi)(3.5)ii,jj,i

134116CHAPTER6.EUCLIDEANKLEINGEOMETRYwheretheparenthesisarounddxj∧dximakesreferencetosummationonlyoverelementsofabasis(Ifdx1∧dx2isintheexpansion,dx2∧dx1isnot).Thisisacovariantexpression,inthesensethattheexteriorderivativeofthesamedifferential1−formgivenintermsofanothercoordinatesystem,vdyi,yieldsithesamedifferential2−formdα:(v−v)(dxj∧dxi)=(v−v)(dyj∧dyi).(3.6)i,jj,ii,jj,iInotherwords,dαiscoordinateandframefieldindependent.Itscomponents(i.e.thecontentsofthefirstparenthesisoneachsideof(3.6)),transformlin-earlybecausesodotheelementsofthebasisofdifferential2−forms.(3.4)isconnectiondependent;(3.5)isnot.Oneshouldbeawareofthefactthatsometimesexpressionsshowtheconnec-tionexplicitly,andyettheydonotdependonit.Forexample,inlaterchaptersweshalldealwiththesocalledLevi-Civitaconnection,forwhichthefollowingiscorrect:dα=vωj∧ωi+vdωk=(v+vΓk)ωj∧ωi.(3.7)i/jki/jkjiThepresenceofthegammasisoftenanindicationofdependenceofconnection.Butnotalways,(3.7)beinganexample.Thefirstequationofstructure,dωk=ωj∧ωkdoesnotimplythatdωkisconnectiondependentsinceitisdefinedjwithoutresorttoit.Thisequationsimplystatesthatconnectionsthatsatisfyithavezerotorsion.Thespuriousdependenceonconnectionpropagatestoequationswhereithasbeenusedtoreplacedωkwithωj∧ωk.j6.4HodgedualityandstaroperatorThetopicofHodgedualityisbesttreatedusingCliffordalgebra.Thatwillbedealtwithinanotherbookbeingpreparedbythisauthor.Inthissection,wesimplypresentsomebasicresultsofthatalgebra,specificallyHodgeduality,withoutresortingtoaformalpresentationofthesame.Givenadifferentialr−form,wedefineitsHodgedual(orsimplydual,whenthereisnoambiguity)asthedifferential(n−r)-formthatoneobtainsbyitsCliffordproductwiththeunitdifferentialn−form.Intheinterestofexpediency,weproceedtoexplainthisconceptwithexamplesandtheeasycomputationsthatresultwhenoneexpressesdifferentialformsasproductsofthedifferen-tial1−formsdx,dy,dz,dr,rdθ,rsinθdφ,dρ,ρdφ,dzassociatedwithorthonor-maltangentvectorbasisfields,specifically(i,j,k),(er,eθ,eφ)and(eρ,eφ,ez).3inξ.Weshallrefertothosebasesofdifferential1−formsthemselvesasor-thonormal.Thenotationwillbe(ωi)inthreedimensions,and(ωμ)inotherdimensions.3Inξ,weusethesymbolwtorefertotheunitdifferential3−form,i.e.w=ω1∧ω2∧ω3=dx∧dy∧dz=dr∧rdθ∧rsinθdφ=...(4.1)

1356.4.HODGEDUALITYANDSTAROPERATOR117Asimplerearrangementoffactorsyieldsw=r2sinθdr∧dθ∧dφ=g1/2dr∧dθ∧dφ,(4.2)wheregisthedeterminantofthematrixmadewiththecoefficientsofthemetric.Weshallshowattheendofthissection(Eq.(4.25))thegeneralityofthistypeofresult.Consider,forinstance,(dx∧dy)∨w,i.e.(dx∧dy)∨(dx∧dy∧dz).Whenthereisorthonormality,dx∧dyequalsdx∨dy,anddx∧dy∧dzequalsdx∨dy∨dz.Tominimizeclutter,weusejuxtapositioninsteadofthesymbol∨,andshallalsorepresentα∨was∗α,whichiscommonintheliterature.Thenotation∗dx∧dywillbeviewedassignifying∗(dx∧dy)ratherthat(∗dx)∧dy.Wethenhave∗dx∧dy=(dxdy)(dxdydz)=−dxdydydxdz=−dz,(4.3)wherewehaveusedtheassociativepropertyandthatdxdy=−dydx,dxdx=dydy=dzdz=1.(4.4)Letusobservethatw∨w=dxdydzdxdydz=dxdxdydzdydz=dydzdydz=−dydydzdz=−1.(4.5)Also,using(4.3)and(4.5),weget∗dz=∗(−∗dx∧dy)=−∗(∗dx∧dy)=−dxdyww=dx∨dy,(4.6)andsimilarly,bycyclicpermutationof(4.6),∗dx=dy∧dz,∗dy=dz∧dx(4.7)and∗∗dy∧dz=−dx,dz∧dx=−dy.(4.8)Finally,∗1=dx∧dy∧dz,(4.9)∗dx∧dy∧dz=−1,(4.10)and∗fdx∧dy∧dz=−f,(4.11)byvirtueof(fα)w=f(αw).(4.12)Clearly∗∗α=−α.Letω1,ω2,ω3beabasisofdifferential1−formsdualtoanorthonormalbasisfield.Momentarily,andinordertoemphasizethattheequationsabouttofollowareonlyvalidforωi’sdualtoorthonormalbases,weshallusethecircumflex,iω.Wethushave∗123ω∧ω∧ω=−1.(4.13)

136118CHAPTER6.EUCLIDEANKLEINGEOMETRYInparalleltoequations(4.3)-(4.10),wehave∗ω2∧ω3=−ω1,∗ω3∧ω1=−ω2,∗ω1∧ω2=−ω3,(4.14)∗ω1=ω2∧ω3,∗ω2=ω3∧ω1,∗ω3=ω1∧ω2,(4.15)∗1231=ω∧ω∧ω,(4.16)∗123ω∧ω∧ω=−1.(4.17)Wedefinetheunitn-volumedifferentialformas12nω∧ω∧...∧ω.(4.18)Wewishtoexpressitintermsofcoordinatebasesofdifferentialforms.Theseiarenotorthonormalingeneral.Wereplacetheωwiththeirexpressionsintermsofarbitrarycoordinatebasesofdifferential1−forms,i.e.ωμ=Aμdxν,(4.19)νthusobtainingω1∧ω2∧...∧ωn=A1A2...Andxλ1dxλ2∧...∧dxλn.(4.20)λ1λ2λnAlltheindicesλaredifferentandallproductsareequaltodx1∧dx2∧...∧dxniuptosign.ThecoefficientofthisfactoristhedeterminantAofthematrixoftheAν.Wethushaveμω1∧ω2∧...∧ωn=|A|dx1∧dx2∧...∧dxn.(4.21)iwhere|A|istheabsolutevalueofA(incasetheorientationofthebasesωanddxiweredifferent).ItisnotcustomarytoworkwithA.Oneratherusesitsrelationtothemetric,whichweproceedtoderive.Leteandebedualtoωμanddxμμμrespectively.Wewhenhaveg=e·e=AλAπe·e=AλgAπ.(4.22)μνμνμνλπμλπνIntermsofmatrices,thisiswrittenasg=AgA=AA(4.23)whereAisthetransposeofA.Takingdeterminants,wehave1/2|A|=|g|,(4.24)and,therefore,ω1∧ω2∧...∧ωn=|g|1/2dx1∧dx2∧...∧dxn,(4.25)whichisthesoughtresult.Whenthemetricisnotpositivedefinite,themostinterestingmetricisspace-time’s.Wehavetheoptionofchoosingthesignatures(-1,1,1,1)and(1,-1,-1,-1).Inbothcases,(dtdxdydz)2=−1,andwecanchoosetodefinethedualthroughmultiplicationwithdt∨dx∨dy∨dz,orwithdx∨dy∨dz∨dt(=−dt∨dx∨dy∨dx).

1376.5.THELAPLACIAN1196.5TheLaplacianInthisauthor’sopinion,K¨ahler’scalculusisthebesttooltodealwithLapla-cians.Itinvolvesanoperator,calledtheDiracoperator(notexclusivetothatcalculus),thatencompassesthecurlandthedivergenceandisapplicabletodifferentialformsofarbitrarygradeandarbitraryvaluedness.Letusdenoteitas∂.Adifferentialform,α,issaidtobestrictlyharmonicifandonlyif∂α=0.(5.1)Intheparticularcaseofscalar-valueddifferential1−formα,weobtainwhat,intermsofcomponents,areknownasthecurlanddivergenceofavectorfieldwiththesamecomponentsasthedifferential1−form.TheLaplacianofαisdefinedas∂∂α.For0−forms,∂∂istheusualLaplacianoperator.Adifferentialformiscalledharmonicif∂∂α=0.(5.2)K¨ahler’smostelegantapproachto∂and∂∂willbeconsideredonlyinthenextbook,presentedinthelastchapterofthepresentbook.Consequently,weshallnotapproachheretheLaplacianoperatoras∂∂.Luckily,∂∂becomessimply“−∗d∗dα”inEuclideanspacesandonmanifoldswithEuclideanconnectionswithzerotorsion(calledLevi-Civitaconnections).iWeshallagainconsiderbasesωdualtoorthonormalbasisfields,i.e.suchthatiidP=dAai=ωei.(5.3)LetusexaminefromthissimplifiedperspectivetheLaplacianofascalarfunctionfin3Dintermsoforthonormalcoordinatesystems.TheHodgedualofidf=f/iω.(5.4)is∗jkdf=f/iω∧ω,(5.5)withsummationovercyclicpermutations,writingthen(5.5)asinEq.(6.4)ofchapter4,differentiatingit,obtainingthedualandchangingsign,weget∂(fh2h3)∂(fh3h1)∂(fh1h2)1/1/2/3++.(5.6)h1h2h2∂x1∂x2∂x3Theexpression(5.6)isknowninthevectorcalculusastheLaplacianofthefunctionfintermsofortogonalcoordinatesystems.IntermsofCartesiancoordinates(h1=h2=h3=1),(5.6)reducesto∂2f∂2f∂2f++.(5.7)∂x2∂y2∂z2

138120CHAPTER6.EUCLIDEANKLEINGEOMETRYExercise.Use(5.6)tocomputetheLaplacianinsphericalandcylindricalcoordinates.Insphericalcoordinates,wehaveds2=dr2+r2dθ2+r2sin2θdφ2,(5.8)and,therefore,123ω=dr,ω=rdθ,ω=rsinθdφ.(5.9)Wesimilarlyhave,forcylindricalcoordinates,123ω=dρ,ω=ρdφ,ω=dz.(5.10)Thedefinition−∗d∗dfdoesnotrequiretheuseofbasesωidualtoorthonor-malfieldsoftangentvectorbases.Wehaveusedtheminordertoconnectwithformulasusedbyelectricalengineers.WeshallnowobtaintheLaplacianinlessspecializedcasesfortheexclusivepurposeofconnectingwithpractitionersoftensorcalculus,whosetangentbasisfieldsarenotorthogonal.Again,readersareinformedthatitwouldbemuchsimplertodealwith∂∂ineachparticularcasewheninpossessionofsomebasicknowledgeoftheK¨ahlercalculus.TheHodgedualofthe“gradient”ofascalarfieldfis∗fdxl.is,l∗1/2l1ndf=f,l|g|dx(dx∧...∧dx)(5.11)wherewehaveused(4.25).Inthenextbook,weshallshowthatndxl(dx1∧...∧dxn)=(−1)l−1glmdx1∧...∧dxm∧...∧dxn,(5.12)l=1wheretheoverbarmeansthatthefactordxmismissing.Thus(5.11)canbewrittenfurtheras(−1)l−1f|g|1/2glmdx1∧...∧dxm∧...∧dxn.(5.13),lWhenweexteriordifferentiatethisexpression,themthtermonlycontributeswithdifferentiationwithrespecttoxm.Thus∗1/2lm1nddf=(f,l|g|g),mdx∧...∧dx,(5.14)wherethefactor(−1)l−1disappearsbecauseithasbeenusedtomovedxmfromthefronttoitsnaturalposition.Usingagain(4.25),wehaved∗df=|g|−1/2(f|g|1/2gln)ω1∧...∧ωn(5.15),l,mand,therefore,∗∗−1/21/2lm−ddf=|g|(f,l|g|g),m.(5.16)ThisistheLaplacianthatonefindsinbooksonthetensorcalculus,whichwehaveobtainedwithouttheuseoftensors.Exercise.InE3,relatethisto(5.6)whenthesystemofcoordinatesisorthog-onal.Hintglmthenisadiagonalmatrix.

1396.6.EUCLIDEANSTRUCTUREANDINTEGRABILITY1216.6EuclideanstructureandintegrabilityFromnowon,wedropthecircumflex.Sinceweknowwhattheaffineequationsofstructuremeaninaffinespace,andsinceweknowwhattherelationbetweentheaffineandEuclideanspacesis,itisclearthattheequationsofstructureofthesespacesarethesamedωμ−ων∧ωμ=0,(6.1)ννλνdωμ−ωμ∧ωλ=0,(6.2)butaugmentedbytherestrictionωμν+ωνμ=0,(6.3)intheEuclideancase.Equation(6.3)isthedifferentialformofthestatementthatthebasesofEuclideanframebundles,thustobecalledEuclideanbases,areorthonormal.ThesystemofequationsdP=ωμe,de=ωνe,(6.4)μμμνforsomegivensetofdifferentialformsisintegrableifandonlyiftheequations(6.1)-(6.2)aresatisfied.Inthatcase,theintegrationyieldsthePandeμofaffinespace,andofEuclideanspaceinparticular(i.e.when(6.3)isaddedtothesystemtobeintegrated).Wemaybegivenasetofdifferentialformsofthetype(6.3)intermsofsomeunknownsystemofcoordinatesandwewanttoknowwhethertheybelongornottoaEuclideanspace(meaningtoitsframebundleandsectionsthereof).Thetestliesincheckingwhethertheysatisfyornotthe(6.1)-(6.2)system.Iftheanswerispositiveand,inaddition,Eq.(6.3)issatisfied,thedifferentialforms(ωμ,ωλ)willbelongtotheframebundleofaEuclideanspace.νIngeneral,theseformsmaylookverycomplicated.WhenusingCartesiancoordinates,Xμtheytakearathersimpleform,μμ−1νκμ−1λω=dX(R)μ,ων=dRν(R)μ,(6.5)asspecializationsorrestrictionsthattheseequationsareoftheinvariantformsofaffinespace.HereRisthegeneralmatrixoftheorthogonalgroupforthegivennumberofdimensions.Giventhateventhepull-backstosectionsoftheinvariantformsmaytakeverycumbersomeforms,oneneedsanintegrabilitytesttoascertaintheirbelongingtoaffinespace(Euclideanspaceif(6.3)issatisfied).Asonecouldexpect(weshallnotgiveaproof,thepull-backoftheexteriorderivativeequalstheexteriorderivativeofthepull-back.Thisimpliesthattheaforementionedtestofintegrabilitycanbeusedalsointhesections,whichiswhereoneworksmostofthetime.Wehavealreadyseensincethepreviouschapterwhatthoseintegrabilityconditionsareforaffinespace.Letusnowsayalittlebitmoreaboutintegrabilityconditionsingeneral.

140122CHAPTER6.EUCLIDEANKLEINGEOMETRYConsidertheverysimplequestionofwhethertheequationdF(x,y)=f(x,y)dx+g(x,y)dy(6.6)isintegrableornot.Allreadersknowtheanswer,butwewanttoexposethemwithasimpleexampletoFrobeniustestofintegrability.Wefirstwrite(6.6)asdF−fdx−gdy=0,(6.7)andthenseewhethertheexteriorderivativeofthelefthandsidevanishesiden-ticallyornot.Forthispurpose,onemayuse,ifneeded,theequations(s)itself(themselves)whoseintegrabilityisbeingconsidered.Inapplyingthetheorem,oneassumesthatthesolution(Finthiscase)exists,andreachesacontradiction(notgettingidenticallyzero)ifitdoesnotexist.WewouldthuswriteddF=0,andthedifferentiationofthelefthandsideof(6.6)wouldsimplyyield0−f,ydy∧dx−g,xdx∧dy,(6.8)whichdoesnotvanishidenticallyunlessf,y=g,x.Letuslookatthesameprobleminaslightlydifferentway.ConsiderthesystemdF=fdx+gdy,f,y=g,x.(6.9)Now(6.8)isidenticallyzerousingtheequationf,y=g,xofthesystem,whichis,therefore,integrable.The(ingeneral)notintegrableequation(6.7)yieldstheintegrablesystem(6.9)byaddingtoititsintegrabilityconditions.Whathasjustbeensaidappliesequallywelltosystemsofdifferentialequa-tionsinvolvingvector-valueddifferentialforms.ThedifferentiationofthelefthandsideoftheequationsdP−ωie=0,de−ωie=0,(6.10)ijjiyields(−dωi+ωj∧ωi)e,(6.11)jiiki(−dωj+ωj∧ωk)ei,(6.12)afterusingtheequationsde=ωie,whichmakespartofthesystem(6.10).jjiFrobeniustheoremthusstatesthattheintegrabilityconditionsfor(6.10)aregivenbytheequationsobtainedbyequatingtozerotheexpressions(6.11)and(6.12).IntheEuclideancase,wealsohavetodifferentiateωij+ωji,whichyields,using6.3itself,dωij+dωji=dωij+d(−ωij),(6.13)whichisidenticallyzero.Hencetheintegrabilityconditionsarethensimplythesameonesasintheaffinecase.ThisimplicationwouldhavebeenthesameifwetookintoaccountthesocalledaffineextensionoftheEuclideanconnection.Inthiscase,Eq.(6.4)isreplacedwithdg+ωkg+ωkg=dg+ω+ω=0(6.14)ijikjjkiijijji

1416.7.THELIEALGEBRAOFTHEEUCLIDEANGROUP123fortheaffineextensionoftheEuclideanframebundle.Asifitwereneeded,oneeasilyverifiesthatthisequationisintegrable.Exercise.Givenω1=dx1,ω2=x1dx2andω2=dx2=−ω2,findifthese21differentialformsbelongornottoasectionoftheframebundleofEuclideanspace(Theydo!).Exercise.Dothesameforω1=dx1,ω2=sinx1dx2,ωj=0(Theydonot!).iIfthesystemofconnectionequationsisnotintegrable,onecanstillintegratethemoncurvesbetweenanytwogivenpoints.Theresultwilldependonpath.The“P’s”resultingfromtheintegrationofωiealongtwodifferentcurvesibetweenthesametwopointswillbedifferentinthecaseofω1=dx1,ω2=sinx1dx2,ωj=0,eventhroughthesubsystemde=ωje=0isobviouslyiiijintegrable.WewouldnolongerbeinE2.6.7TheLiealgebraoftheEuclideangroupExceptforthefactthattheroleoftheaffineandlineargroupsisnowtakenbytheEuclideanandorthogonalgroups,whatwassaidinsection5.9appliesalsohere.Aswasthecasethen,wehave0ωii,(7.1)0ωjwithωj=−ωiinthepositivedefinitecase.Insteadof(9.6)ofthepreviousijchapter,wenowhaveωkα+ωiβj,(7.2)kjiwhereβi=αj−αi,andwithsummationover,say,i

142124CHAPTER6.EUCLIDEANKLEINGEOMETRYOnereadilychecksthattheβmatricesconstituteanalgebraunderskew-symmetrizedmatrixmultiplication,thoughnotundersimplymatrixmultipli-cation.Thus:23321β1β2−β2β1=β3,(7.6)andcyclicpermutationsthereof.Thesamerelationsapplytotheunprimedbetas.6.8Scalar-valuedclifforms:K¨ahlercalculusAneglectedareainthestudyofEuclideanspacesisitsK¨ahlercalculus.Startbydefiningitsunderlyingalgebra,calledK¨ahleralgebra.ItislikethetangentCliffordalgebrasofsection8ofchapter3,exceptthatthesigmasorthegammasarenowreplacedbythedifferentialsdxμ.Letαbethedifferentialformνλα=aνλ...dx∧dx∧...(8.1)Defineitscovariantdifferentialdμα[46],[48]as∂aνλ...νλdμα=dx∧dx∧...(8.2)dxμinCartesiancoordinates.Ifαisadifferentialr−form,soisdμα.Thecompo-nentsofdμαinothercoordinatesystemsarenolongerthepartialderivativesofthecomponentsofα.ReadersneedonlyrecalltheirknowledgeofthedivergenceofavectorfieldintermsofarbitrarycoordinatesystemstorealizethatitisnotsimplyamatterofreplacingthepartialderivativeswithrespecttoCartesiancoordinateswithpartialderivativeswithrespecttoarbitrarycoordinates.K¨ahlerdefinesa“derivative”∂αas∂α=dxμ∨dα=dα+δα,(8.3)μhavingdefineddαandδαasdα≡dxμ∧dα,δα≡dxμ·dα.(8.4)μμThesetwodifferentialformsplaytherespectiverolesofcurlanddivergence.Theseconceptsarenowbeingmadetoapplytodifferentialforms,ratherthantovectorfields.K¨ahlercreatedandusedthiscalculustoobtainthefinestructureofthehy-drogenatomwithscalar-valuedclifforms[47].Moreimportantly,heusedthesetoshowhowtheconceptofspinoremergesinsolvingequationsfordifferentialforms,eveniftheequationsarenotrestrictedtotheidealstowhichthespinorsbelong[47].In1962[48],hepresentedthegeneraltheorymorecomprehensivelythanin1960[46],andusedittogetthefinestructuremoreexpeditiouslythanin1961[47].Averyimportantandyetoverlookedresultthatheobtainedis

1436.9.RELATIONBETWEENALGEBRAANDGEOMETRY125thatantiparticles(atleastinhishandlingoftheelectromagneticinteraction)emergewiththesamesignofenergyasparticles[48].In[83],theauthorofthisbookhasshownthatcomputationsinrelativisticquantummechanicswithK¨ahler’scalculusofscalar-valuedclifformsaremucheasierandlesscontrivedthanwithDirac’scalculus.TheK¨ahlercalculusofscalar-valueddifferentialformsistoquantummechanicswhatthecalculusofvector-valueddifferentialformsistodifferentialgeometryand,thus,togeneralrelativity.MissingistheuseinphysicsofaK¨ahler’scalculusofdifferentialformstoarbitraryvaluedness,speciallyCliffordvaluedness.6.9RelationbetweenalgebraandgeometryWeexplainedinsection7ofchapter1thatapointseparatesalgebrafromgeometry.Letussayitagain.Inalgebrathereisazero;ingeometry,thereisnot.Wenowproceedwithadditionaldifferences.InEuclideanvectorspaces,thereistheconceptoforthonormalbases.Butthenon-orthonormalbasesareaslegitimateasthosewhichare.Ontheotherhand,inthegeometryofEuclideanspaces,orthonormalbaseshavealegit-imacythatarbitrarybasesdonot.Why?Becauseingeometrytheroleofgroupsbecomesparamount.EuclideangeometryisthestudyoffiguresandpropertiesthatareinvariantundertheEuclideangroup,thebasicfiguresbeingtheorthonormalbasesatallpoints.Affinegeometryisthestudyoffiguresandpropertiesinvariantundertheaffinegroup,thebasicfiguresbeingallvectorbasesatallpoints.Relatedtowhathasjustbeensaid,groupsG0inthepair(G,G0)arenotgiveninalgebratherelevancetheyhaveingeometry.Onthesamegrounds,onespeaksoffibersandbundlesingeometry;muchlessso,orevennotatall,inalgebra.

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147Chapter7GENERALIZEDGEOMETRYMADESIMPLE7.1OfconnectionsandtopologyThisisatransitionalchapterinthesensethatwestudysimpleexamplesofCartan’sgeneralizationofaffineandEuclideanKleingeometries.Forfurtherperspective,wealsodevoteasectiontotheoriginalRiemanniangeometry,whereRiemann’scurvaturehadtodowithaso-calledproblemandmethodofequiva-lence.“Equivalence”isnotatopicofdifferentialgeometryproper,butmaybeusedtherenevertheless.Weshallstudythreecommonsurfacesundertwodifferentrulestocomparetangentvectorsatdifferentpointsonthosesurfaces.OneoftheserulesisthesocalledLevi-Civitaconnection(LCC).Formulatedin1917,itwasthefirstknownaffineconnection(weshalllatercallitEuclideanratherthanaffine).Itwasadopted,maybeunnecessarily,bygeneralrelativity.Wesay“maybeunnecessarily”becauseeventuallytherewereotherconnectionscompatiblewiththeoriginalRiemanniangeometryandwithgeneralrelativity,butwithbetterproperties.ThemostobviousalternativetotheLCCistheColumbusconnection(seePreface),thereasonforthischoiceofnamelatertobecomeclear.Itisofthetypecalledteleparallel,whichmeansthatonecanthenestablisharelationofgeometricequalityoftangentvectorsatdifferentpointsofaregionofamanifold.Ifthereareexceptionalpoints(evenasfewasjustone)wherewecannotdefinetheequalityofitstangentvectorstothetangentvectorsatotherpointsofM,wesaythattheteleparallelismislocalorlimitedtoregionsofthemanifold.Thetopologyofthemanifoldmayimpededefiningtheconnectioneverywhere,whileallowingitonregions.129

148130CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEConsiderthefollowingeasy-to-understandexampleofalocalproperty.ThedirectionEast(oranyotherdirectionforthatmatter)isnotdefinedatthepoleswhere,correspondingly,theColumbusconnectionisnotdefinedeither.OnecannotputconnectionsotherthantheLCConthefull2-sphere,i.e.thespherein3-dimensionalEuclideanspace.Wethensaythattheyarenotgloballydefined.Itisatopologicalissue.TheColumbusconnectionisgloballydefinedonmanifoldsthatareregionsofa2-sphere,amongthemtheoneresultingfrompuncturingitatthepoles.Puncturingchangesthetopologyofthesphere.Thisdifferenceislargelyaca-demicforthephysicistwhointegrateshisequationtofindtheworldinwhichhelives.Cartanindeedspokeofsystemsofequationswhosesolutionsarethemanifoldsinwhichtheconnectionlives[29].Thusthe2-spherewouldbethesolutiontoasystemofequationsofstructurewiththeLCC,butnotwithanyotherconnection.Assumethat,makingabstractionofdimensionalityandofsignatureofthemetric,thespacetimesolutionofsomehypotheticalsystemofequationswerelikeaspherebutwithColumbusconnection.Itwouldnotbethewholespherebutjustaregionofit.Theenergyrequiredtocreatetheregionwouldbegreaterandgreaterasoneapproachedthefullsphere.Onlyinfiniteenergywouldclosethesurface,sinceitistobeexpectedthatthesingularityofthetorsionatthepoleswouldentailasingularityofrequiredenergy,aswellassingularityofotherphysicalquantitiesrelatedtoaffine(saidbetter,Euclidean)structure.Hence,weshalltaketheconceptofteleparallelismandzeroaffinecurvatureassynonymous,atleastfromaphysicalperspective.TheLCC,ontheotherhand,isgloballydefined.ButtheLCCdoesnotincludeaconceptofgeometricequality.Mostpractitionersofdifferentialge-ometrywouldconsiderittobethenaturalconnectionbecauseitiscanonicallydefinedbythemetric.Butthereis,forexample,thetorus,where,byreasonofsymmetry,theLCCconnectionislessnaturalthanthealwaysricherColumbusconnection.Thedecisionastowhatistheaffineconnectionofspacetimewastakenwhentherewasnotageneraltheoryofconnections,andofteleparallelconnectionsinparticular.Theadoptionmightonedaybeviewedasamistakeinducedbyhistoricalcircumstance.Forthemoment,letusunderstandthemathematics.7.2PlanesThetermplanemeansdifferentthingsindifferentcontexts.Underafirstmean-ing,planeisanylinearspace,notnecessarily2-dimensional.Thiswasthemean-ingofplanewhenthistermwasusedinthedefinitionofaLiealgebra.Wemighthaveusedthetermhyperplane,butwehavefollowedcustominusingthetermplaneinthatcase.Othertimesplanemeansthetwodimensionalaffinespace.Oronemaybereferringtoann−dimensionalstructurewherethesquareofthedistance

1497.2.PLANES131betweentwopointsmaybedefinedbyanexpressionoftheform(x1)2+(x1)2+...+(xn)2,(2.1)evenifacomparisonofvectorsatdifferentpointshasnotbeendefined.Inparticular,thereisthe2−dimensionalmetricplane.TheEuclideanplaneis,aswehaveseen,theaffineplaneendowedadditionallywithasquaredistanceasin(2.1)withn=2.7.2.1TheEuclidean2-planeWereturntotheEuclideanplaneforlatereasycomparisonofitsLLCwiththeColumbusconnection.Recalltheimportantpointthattheconstantbasis(i,j)maybeviewedasaconstantbasisfield,i.e.i(A)=i(B),j(A)=j(B),(2.2)foranytwopointsAandBinthespace.LetusadvancethattheconnectiondefinedinthiswayistheLCC,whichweshalllaterdefine.Itiszerointhisparticularsection,di=0,dj=0,(2.3)butnotinthebundle,aswehavealreadyseen.ConsidernowtheEuclideanbasisfieldintermsofwhichtheelementarydisplacementdPmaybewrittenas:dP=dρeρ+ρdφeφ.(2.4)IfweperformthedotproductofdPwithitself,wegetdP·dP=dρ2+ρ2dφ2,(2.5)wherewehavetakenintoaccountthateρ·eρ=1,eρ·eφ=0,eφ·eφ=1.Itisclearthateρ=cosφi+sinφj,(2.6a)eφ=−sinφi+cosφj.(2.6b)Wedifferentiate(2.6)andobtain:deρ=−sinφdφi+cosφdφj,(2.7a)deφ=−cosφdφi−sinφdφj.(2.7b)Werefer(deρ,deφ)tothebases(eρ,eφ)themselvesandinvert(2.6)toobtainicosφ−sinφeρ=,(2.8)jsinφcosφeφ

150132CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEwhichpermitsustothengetdeρanddeφasdeρ−sinφcosφcosφ−sinφeρ=dφ(2.9)deφ−cosφ−sinφsinφcosφeφfrom(2.7)and,finally,deρ01eρ=dφ,(2.10)deφ−10eφordeρ=dφeφ,deφ=−dφeρ.(2.11)WethusreadfortheLCCoftheEuclideanplaneinthesectionunderconsid-eration:1212ω1=0,ω1=dφ,ω2=−dφ,ω2=0.(2.12)Readersmaywanttotesttheirunderstandingbybuildingfrom(2.12)thediffer-entialformsωjinthebundleoforthonormalbases.WedonotneedtocomputeithetorsionandtheEuclideancurvaturesincewehavenotabandonedatanypointtheEuclideanplaneandtheyare,therefore,zero.7.2.2Post-Klein2-planewithEuclideanmetricWenowintroducetheColumbusconnectiononthemetricplane(puncturedatonepoint),whichgeneratesaCartangeometry.Specifically,thiswillbeapost-KleineanCartangeometry(space,manifold)thatisflatinthemetricsense.Thistypeofflatnessmeansthatthemetriccanbewrittenasasumofsquaresofthedifferentialofthecoordinatesbecausewekeeptheds2=dx2+dy2oftheEuclideanspace.Butweshallonlychangetherelationofequalityofvectorsattachedtodifferentpointsofourmanifold.Wedonotyetneedtounderstandeverything,asallofitwillbecomeobviouslittlebylittle.Supposewehadaflatworldinwhichtherewereanadvancedsocietyof2-dimensionalbeings.SupposefurtherthattherewereapointOontheplaneplayingtheroleofoursun.Barringothercausesfordifferencesinclimate,unlikebetweendifferentpointswiththesamelatitudeontheearth,allpointsatthesamedistancefromOwouldbeequivalentfromaclimatologicalperspective.Therewouldbesometemperateregioninacircularbandwithradiiclosetosomeidealclimateradius,r=r0.Wewouldputthosewhohavetosweattomakealivingatdistancesr>>r0,whereitiscold.Theveryhotregionwherer<

1517.2.PLANES133ratherthan(2.11).Also,sinceweareinthemetricplane,wewouldstillhaveds2=dρ2+ρ2dφ2.(2.14)Saidbetter,wewouldstillhave(2.14)asaconsequenceofdefiningds2asdP·dPwithdPgivenbydP=dρeρ+ρdφeφ,(2.15)and,again,witheρ·eρ=1,eρ·eφ=0,eφ·eφ=1.Forcompleteness,weexpressthisconnectionintermsofthe(i,j)basisfield.DifferentiatingEqs.(2.8),wewouldhave,using(2.13):di−sinφ−cosφeρ=dφ.(2.16)djcosφ−sinφeφTheinversionof(2.8)yieldseρcosφsinφi=.(2.17)eφ−sinφcosφjSubstituting(2.17)in(2.16),wegetdi0−1i=dφ,(2.18)dj10jor,equivalently,di=−dφj,dj=dφi.(2.19)Weprovidesomeremarksforreaderstothinkabout(Asystematicpresen-tationofthetheorythataddressestheissuesinquestionwillbegivenincomingchapters).DefinetheEuclideancurvatureaswedidtheaffinecurvature:jjkjΩi≡dωi−ωi∧ωk.(2.20)WecallitEuclideanratheraffinebecauseitlivesinadifferentbundleandtakesvaluesinadifferentalgebra.ItpertainstoEuclideanconnections,whicharecalledaffineconnections(specificallymetriccompatibleaffineconnections)intheliterature,butonlyveryexceptionallyinCartan’swritings.Weshallreturntothislateron.TheEuclideancurvaturestilliszerointhepuncturedplaneendowedwiththeColumbusconnectionsincethereisequalityofvectorsatadistance,thoughnotinthesamewayasintheprevioussubsection.Definethetorsionalsoasintheaffinecase:iijiΩ≡dω−ω∧ωj.(2.21)ReadersmayverifythatthetorsionisnotzerobutΩ1≡d(dρ)−0=0,Ω2≡d(ρdφ)−0=dρ∧dφ.(2.22)ConsidernextthedevelopmentintheEuclideanplaneofaclosedcurveofthisuniverse.Theradiallinesandconcentriccirclescenteredattheoriginof

152134CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEtheradiallineshaveconstantdirection.Sincethecirclesareperpendiculartotheradii,acurvilinearquadrilateralconstitutedbytheintersectionoftworadiallineswithtwoofthosecirclesdevelopsintoanopencurveinEuclideanspacewhenthefoursegmentsofthecurvilinearquadrilateralgointorectilinearsegmentsinEuclideanspace,thesebeingorthogonaltotheadjacentones.Thiscurvedoesnotclosebecausetheradiallinesinterceptarcsofdifferentsizeonthetwocircles.Inthiscase,thefailuretocloseisthemanifestationofnon-vanishingtorsion.Warning:thedevelopmentofacurvemaynotclosealsoifthetorsionwerezero,butforadifferentreason.Weshallseemoreonthislater.Thereisadifferentwayoflookingatthisfailuretoclose.LetthecurvilinearquadrilateralhaveconsecutiveverticesA,B,C,D.Assumewerepresentontheplanenotjustoneofthefoursegmentsinsuccession,butfirstABCandthenADC.Wecannotassignavectortothepairofpoints(A,C)sincethedevelop-mentsofABCandADCendatdifferentpoints.Equation(2.15)remainsvalid,butavectorPdoesnotexistinsuchamanifoldconsistentlywiththeColumbusrulethatwehaveimposedonit.Coordinates(x,y)maystillbechosenasx=ρcosφ,y=ρsinφ,(2.23)andthemetric(2.14)againbecomesds2=dx2+dy2.(2.24)Non-experiencedreadersarenotexpectedtounderstand,muchlessremem-berand/orbeabletojustify,everythingthathasbeensaidhere.Wearejusttryingtoprovidetheflavoroftheintrincaciesofdifferentialgeometry,intrica-ciesthatarenotaccessibleeitherbythetreatmentsforphysicistsorintheveryformaltreatmentsformathematicians.Asproofofthis,witnessthecontrivedexamplesofdifferentialmanifoldswithtorsionthataregivenintheliterature.7.3The2-sphereTheColumbusconnectionoftheplanemaybealittlebitconfusingbecauseitadmitscoordinateswhosesquareddifferentialsdiagonalizethemetric(aswehavejustseen)andyetitdoesnothaveadisplacementvectordefinedonit(ofwhichtheCartesiancoordinateswouldbeitscomponents).TheColumbusconnectioniseasiertounderstandonthepuncturedsphere.Itmadehistoryin1492.7.3.1TheColumbusconnectiononthepunctured2-sphereIn1924,CartangavetheexampleoftheearthendowedwiththeColumbusconnection[13].Hedidnotusethisname.Hereiswhywedo.StartinghisfirstmayorvoyageWestwardintotheAtlantic,ChristopherColumbusorderedthecaptainsoftheothertwoshipsthataccompaniedhistomaintainthesamedirection,West!Ifwestretchthingsalittlebitandputthingsinmodernterms,

1537.3.THE2-SPHERE135Columbussaidthis:theunitvectorintheWestdirectionisequaltoitselfatanyotherpoint.Thesamecommentappliestoanyotherdirection(wheredefined,thusnotatthepoles).UndertheColumbusconnection,theautoparallelsorlinesofconstantdirectionarethemeridians,theparallelsandtherhumblines(i.e.curvesthatmakeaconstantanglewiththeparallelstheyintersect,whilespiralingtowardsthepoles).Theparallelsandthemeridiansareonlyparticularcasesofrhumblines.Tangentvectorsarevectorsonthetangentplanestothesurfaceofthesphere.TheColumbusconnectionisonewherethereisarelationofequalityofsuchvectorsatanytwodifferentpointsofthepuncturedsphereindependentlyofanycurveusedtocomparethem.Incontrast,theLCConthesphereandinanymanifoldotherthanEuclideanspacesisnotarelationofequality.Weshallseethisinthenextsubsection.Letusrushtoaddthat,fromalocalperspective,thecylinderandtheconeareEuclideanspaces:justcutthem,openthemupandextendthemontheEuclideanplane.TheLCCandtheColumbusconnectionsarecalledEuclideanconnections.TheypertaintodirectCartaniangeneralizationsofEuclideanspace,notofaffinespace.Thetransformationsinthefibersoftheirbundlesofframesarerotations.Thustheypreserveangles.Weshallusethisfacthere,butshallleavethediscussionoftheseissuesforfuturechapters.TherepresentationinEuclideanspaceofcurvilinearquadrilateralsformedbytheintersectionofpairsofmeridiansandofparallelsfailstocloseingeneral,andthusfailstogivewhatwouldotherwisebearectangle.Intheprevioussubsection,thiswasduetothefactthatthesegmentsofparallelswillingeneralbeofdifferentlength.WearenowreadytowritedowntheColumbusconnectiononthesphere.If(eθ,eφ)areunitvectorsinthedirectionofthe(θ,φ)coordinatelines,wehavedeθ=0,deφ=0.(3.1)Anotheralternativefieldwouldbeone(eθ,eφ)associatedwiththesphericalcoordinates(θ,φ)basedonadifferentpairofpoles.Theequationoftheconnection(3.1)intermsofthenewbasisfieldwouldberathercumbersomewithoutbettermathematicalmachinerythantheoneusedinthebook.Ofcoursethereisaconnectionthattakesasimpleformintermsof(θ,φ),namelyde=0,de=0.(3.2)θφTheColumbusruletocomparedirectionsonthe2-spherebecomesaconnec-tionwhenitisputtogetherwithaconservationofdistances(astickwascon-sideredequaltoitselfwhereveritwent).ItwasnotthoughttobeaEuclideanconnectionbecausetheconceptdidnotexistinthemathematicalliteratureuntilCartanpublisheditin1924[13].Thereareothertypesofconnectionsonthespherewhichalsoareteleparallel,i.ewhereapath-independentequalityoftangentvectorsatdifferentpointsisdefined.Beforeattemptingtobuildthemonthesphere,readersshouldattempttodosoontheplane.Hint:Thinkofadifferentsystemofcoordinatelinesanddefinethemaslinesofconstant

154136CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEdirection.Theydonotneedtobeorthogonalcoordinatesystemsinordertodefineaconnection;theunittangentvectorstothecoordinatelineswillnotthenconstituteEuclidean(i.e.orthonormal)bases.Exercise.ComputethetorsionoftheColumbusconnectiononthepuncturedsphere.Donotworryifyoucannot.Allthiswillbedonesystematicallyincomingchapters.7.3.2TheLevi-Civitaconnectiononthe2-sphereWearenotgoingtobedistractedinthissubsectionwithhowtheLCCisdefinedingeneral.Sufficetosayatthispointthat,accordingtotheLCC,themaximumcirclesandonlytheyarethelinesofconstantdirectiononthesphere.MadridandNewYorkareapproximatelyonthesameparallelof40.50Northlatitude.TheyhaveWestlongitudesof40and740respectively.Whencommer-cialplanesflyfromMadridtoNewYork,theyfollowamaximumcircle.TheythusfollowalineofconstantdirectionoftheLCC.Forthatpurposeandaf-terleavingtheproximityofMadrid’sairport,theyflyindirectiont,whichis,say,DdegreesNorthofWest,andarriveintheproximityofNYinadirec-tiontwhichisDdegreesSouthofWestandwhichtheLCCconsiderstobethesameastfromtheperspectiveofthatmaximumcircle.SinceWestinMadridisDdegreescounterclockwisefromt,thecorresponding(i.e.equal)di-rectioninNYisDdegreescounterclockwisefromtand,therefore,2DdegreescounterclockwisefromNY’sWestdirection.WestinMadridandNYarenotequalbytheLCC,fromtheperspectiveofthemaximumcirclethattheyjointlydetermine.Letusconsidernowanexamplethatisevenmoreextreme.Considertwoantipodalpointsontheequator.WestartgoingNorthonthemeridianfromoneofthosepoints,passthepoleandcontinueonthesamemeridian.WereachtheantipodegoingSouth.TheNorthdirectionhasbecometheSouthdirectiononitsvoyage.Exceptintheplane,whereitistrivial,theLCChastobecomputed.Onecomputesitbysolvingthesystemconstitutedbythestatementthatitstorsioniszero,dωi−ωj∧ωi=0,(3.3)jandthattheconnectionisEuclidean,i.e.ωij+ωji=0,whichinturnimpliesijωj=−ωi.(3.4)jThesystemofequations(3-3)-(3-4)canbesolvedforω.Weshallentertheigeneralmethodofsolutioninafuturechapter.For2-dimensionalEuclideanconnections,thesolvingisveryeasy,actuallybyinspection.Inthecaseofthesphere,wegivethesolutionintheformdeθ=cosθdφeφ,deφ=−cosθdφeθ,(3.5)soastonotforgetitsgeometricsignificance.

1557.3.THE2-SPHERE137Exercise.ShowthatΩ2=−sinθdθ∧dφ=−Ω1.Again,donotworryifyou12cannot.Allthiswillbedonesystematicallyincomingchapters.7.3.3Comparisonofconnectionsonthe2-sphereInthefigure,thecurvilineartrianglePJNisconstitutedby900arcsofmax-imumcircles.TheyhavebeenchosensothatthosearcsarelinesofconstantdirectionbothaccordingtotheLCCandColumbusconnection.UsingtheLCC,Nm........................n..................................m........J..............n..............................................................P....................0...90.................Figure5:TransportofnfromPtoNalongtwodifferentpathslettheunitvectornatPbe“transported”toN,whereitbecomestheunitvectornperpendiculartotheplaneofthepaperandgoingin.Wewouldthenbetemptedtowriten=n,andsimilarlym=m.Ifwewriten=masamatterofdefinition,alltheseequalitieswouldthenimplyn=m,whichisobviouslywrongsincetheymakeanangleof900.Forreasonssuchasthis,Levi-Civitacouldnotwritealltheseequalities.Buthecouldsaythatnistheresultofparallel-transportingwithhisrulenfromPtoNalongthemeridianandthatmistheresultofparalleltransportingnfromPtoNalongthelinePJN.Theresultofthetransportisafunctionofthepathfollowedwhenoneusesthisconnection.AssumenowtheColumbusconnection.Wearenolongertransportingany-thing.SincewecannotdefinedirectionsatNwithoutresorttobasisfieldsassociatedwiththespherepuncturedatthepoles,weconsideraverysmallparallelneartheNorthpoleundertheassumptionthattheearthisaperfectsphere.LetmandnbetheunitvectorspointingNorthattheintersectionofthatparallelwiththemeridiansthroughJandP.Wehavem=m=n=n,(3.6)

156138CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEiftheruletocomparevectorsistheColumbusconnection.WehaveintroducedthatsmallparallelforcomparisonpurposeswiththeLCCastowhathappenswhenwegetarbitrarilyclosetothepole,whichwedonext.ConsidertherepresentationofthesphericaltriangleNJPthattheLCCyieldsontheEuclideanplane.SinceNJandNPareperpendiculartoJP,therepresentationdoesnotclose.Ittakestheform.InthecaseoftheColumbusconnection,wedonothaveasphericaltriangle,butacurvilinearquadrilateralwithJandPasverticesofoneside,theoppositesidebeingarbitrarilysmall.Theresultisalsothesame,exceptforanadditionaltinyhorizontalcontributionatthetoppertainingtothatverysmallparallel.Itisclearthatifthetorsiondoesnotvanish,thefailuretocloseoftherepresentationonaplaneofaclosedcurveonthesphereneednotsoundstrange.Thisisclearsincetorsionisdirectlyrelatedtothetranslationelement.Butitislessclearwhythiswouldalsohappenwhenthetorsioniszeroandthecurvatureisnot,sincecurvatureisdirectlyrelatedonlytothecomparisonofvectors.Thereasonisnototherthanthedifferentvectorsthatweassociatewiththesidesofcurvilinearsegmentsbelongtodifferentvectorspaceswhentheaffine(respectivelyEuclidean)curvatureisnotzero.TheLCCdoesnotpermitonetoidentifytheEuclideantangentvectorspacesatdifferentpoints.7.4The2-torusInthisbook,thetoruswillalwaysmeanthe2-dimensionalorordinarytorus.ItsrelevancevisavisthetheoryofconnectionsliesinthattheColumbusconnectiononthetorusismorenaturalthanonthesphere,andalsomorenaturalthantheLCC.Weshallcallitthecanonicalconnectionofthetorus.Anyplanethroughtheaxisofthetorusintersectsthesurfaceofthisfigurealongtwocirclesofequalradius.Thesecirclesweshallcallmeridians.Aperpendicularplanetotheaxisintersectthetorusalongtwocirclesthatweshallcallparallels.Thesetwoparallelsapproacheachotherastheperpendicularplanemovesawayfromtheequatorialplane,untilthetwobecomejustone.Furtheraway,theplanesnolongerintersectthetorus.7.4.1Canonicalconnectionofthe2-torusByreasonofsymmetry,weshouldconsidertheparallelsandmeridiansofthetorusaslinesofconstantdirection.Alsolinesofconstantdirectionwouldbetherhumblines,meaninglinesthatintersecttheparallelsataconstantangle.Aunitvectoralongarhumblinewouldbeconsideredtobeequaltoitself.Actually,allunitvectorsmakingthesameanglewiththeparallelswouldbeconsideredtobeequal.ThiswouldbetheColumbusconnectiononthetorus.Thisarelationofvectorequalityonthisfigure.Wedonotneedtoremoveanypointsince,unlikethecaseofthesphere,thisequalityofvectorsisdefinedeverywhere.

1577.4.THE2-TORUS139Letuscomputethetorsionofthisconnection.Itwillbegivenbyjustdωi.Letrbetheradiusofthemeridians,andletRbethedistancefromthecenterofthetorus(i.e.thecentertoitsequatorialparallels)tothecentersofthemeridians.Letθandφbeangularcoordinatesonmeridiansandparallelsrespectively.AswasthecasewiththeColumbusconnectiononthesphere,wehavedeθ=0,deφ=0.(4.1)Thedifferentialformsonthissectionoftheframebundleofthetorusareω1=(R+rcosθ)dφ,ω2=rdθ.(4.2)ThetorsionoftheColumbusconnectionisΩ1≡dω1=−rsinθdθ∧dφ,Ω2≡0.(4.3)TheEuclideancurvatureisofcoursezero.Considerthecurvilinearquadrilateraldeterminedbytheintersectionoftwomeridiansandtwoparallelsofdifferentradii.ItsrepresentationinEuclideanspaceagainisaquasi-rectangle.Itfailstoclose.ItisareflectionofthefactthatthetorsionoftheColumbusconnectionisnotzero.Weshalllearnlaterthattheautoparallelsorlinesofconstantdirectionandthestationarycurves(inactualpractice,curvessuchthattheirlengthissmallerthanthelengthofneighboringcurvesbetweenthesametwopoints)donotcoincidein2-dimensionalsurfacesunderdifferentconnections.Thisisalsothecaseinhigherdimensionsingeneral,butnotalways.TheColumbusconnectionisteleparallelsinceanequalityofvectorsatadistanceisdefined.ItcannotbeequaltotheLCCofthetorussincetheLCCsatisfiesthatitslinesofconstantdirectioncoincidewithitsstationarycurves.Thesecurvesarenotingeneraltheparallelsandthemeridians.Weshallnowshowthiswithoutresorttocalculation.Inoneoftheparallelsofthe“upper”semi-torus,sticktwoneedlesseparatedsay,60or90degrees,orsomeothernumberinthatgeneralinterval(forvisualpurposes).Runalengthofchainbetweenthetwoneedles,looseenoughtoachievethatthechainsitsexactlyontheparallel.Onecanpullthechainfromoneendwhilekeepingthechainfixedattheotherend.Thechainmovesup.Thisshowsthattherearecurvesintheneighborhoodofasegmentofparallelbetweenthesametwopointsthathaveasmallerdistance,andthusarenotstationaryattheparallel.Tomakethepointevenmoreevidentbycomparison,considerthesamementalexercisewithtwoneedlesonthesphereandthechainlyingonthemaximumcirclethroughthetwoneedles.Onecannotpullthechain,sinceothercurvesintheneighborhoodhavegreaterlengthbetweenthesametwopoints.WehavedescribedtheColumbusconnectiononthetorus.Itsatisfiesthatitiszerointheconstantbasisfieldjustmentioned.WhatistheLCCofthetorus?Itishighlynontrivial,incontrasttotheColumbusconnection.ThisisthereasonwhywesaidoftheColumbusconnection,whichistrivialandrespectsthesymmetryofthetorus,thatitisitscanonicalconnection.

158140CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLE7.4.2Canonicalconnectionofthemetricofthe2-torusConsidernowtheLevi-Civitaconnection.Usingagainthesystem(3.3)-(3.4),onereadilyfinds21ω1=sinθdφ=−ω2.(4.4)Noticethesimilarityofthisconnectionwiththeconnectiononthesphere(makesomechangesinnotationtomakethesimilaritymoreobvious).Thinkwhatwouldhappenifyoumadetheholeofthetorussmallerandsmaller,untilr=R.Continuethisprocess,makingR

1597.6.USEANDMISUSEOFLEVI-CIVITA141forarbitraryλ,ratherthanω1=dθandω2=sinθdφ.Theωi’sin(5.4)aresuchthatthesumoftheirsquaresyields(5.2),whichisindependentofλ.Letusnowconsiderthisproblemofequivalenceinarbitrarydimension.Cartanremarksthat,theωibeingwhattheyare,thedωimustbeexteriorproductsoftheformdωi=ωj∧σi,(5.5)jforsomeundefineddifferentialforms,σi,whichwilldependonboththeωlandjthedifferentialsofthecoordinatesinthefibers,likethepreviousdλ.Heshowsthattheymustsatisfyσij+σij=0,(5.6)asaconsequenceofthefactthatthedifferentiatedds2mustnotinvolvethedλ.Thedifferentiationofds2mustbecompatiblewiththestructuretowhichds2belongs,whichisnotaexterioralgebrastructure.Thesystem(5.5)-(5.6)canbesolvedforσij.Weshalldealwiththissystematlengthinalaterchapter.Atthispoint,letuscontinuewithCartan’sargu-ment.Hedevelopsfurthertheconsequencesoftheindependenceofds2onthecoordinatesλandfindstheequationdσj=σl∧σj+Rjωl∧ωm(5.7)iililmThismeansthatallthedependenceofdσjontheσm’sandthusondifferentialsiklikethedλ’siscontainedinthetermσl∧σj.TheRjωl∧ωmconstitutetheililmRiemanniancurvature.Therearenoωi∧σjterms,andnoadditionalσl∧σjlilterms.Cartanfinallyconsiderstheintegrabilityofdxi−ωi=0whenaccompaniedbytheintegrabilityconditionσi=0.OnehastoapplytheFrobeniustestofjintegrabilitytothesystemiiidx−ω=0,σj=0.(5.8)Ifσiiszero,dσialsoiszero.Butthisisnotthewayinwhichthetheoremisjjtobeused.Onehastocheckwhetherdσjiszeroin(5.7)whenusingσi=0.ijItisclearthatitisnotingeneral,unlesstheRiemanniancurvatureiszero.AsforusingtheFrobeniusteston(5.8),wehaved(dxi−ωi)=−dωi=−ωj∧σi=0,(5.9)jbyvirtueofthe(partially)definingrelation(5.5)ofσiandthelastequationinjthesystem(5.8).Itthusfollowsthatanecessaryandsufficientconditionforachangeofcoordinatestoexistthatreducethemetrictotheform(5.1)isthattheRiemanniancurvaturebezero.7.6UseandmisuseofLevi-CivitaWenowwishtodiscussthesignificanceofthespecificformofthesystem(5.5)-(5.6)thatdefinesσi,systemwhichisinstrumentalinobtainingtheRiemannianj

160142CHAPTER7.GENERALIZEDGEOMETRYMADESIMPLEcurvaturethrough(5.7).Thatsystemisthesameoneas(3.3)-(3.4)forωi.Thejσjmustthusbeidentifiedwiththeωi,butonlyintheformthatthesolutionsijtotwodifferentproblemstake.Theyarerepresentationsofthesolutionofjtwodifferentproblems.Theσisolutionwasnotaconnection(ofbases)before1917,sincemanifoldsotherthantheaffineandEuclideanspacesthemselveshadnotbeenendowedwithsomesortofcomparisonoftangentvectorsatdifferentpoints.jDuringthelastthirdofthenineteenthcentury,thecomponentsofσiwereknownunderthenameofChristoffelsymbols,combinationsofthemetricanditsfirstorderderivatives.In1917,Levi-Civita(LC)conceivedthepossibilityoftheiruseforsuchcomparisons[52](notatrueconceptofequalitybutasucceda-neumforit).Thiswasequivalenttousingσifortherolethatωihasplayedjjinthisandthepreviouschapter.Theconceptofωionarbitrarydifferentialjmanifoldshadnotexisteduntilthen;theconceptsofaffineandEuclideanspaceexisted,butmuchofwhathasbeenstatedinthisbookaboutthosespaceswasnotyetdeveloped.Manywhoshouldknowbetterstillfailtounderstandwhatexactlyhappenedin1917,partlybecausetheyarenotawareofhowRiemanniangeometryemergedandgrew.TheLevi-Civitaconnectionemergedintheabsenceofageneraltheoryofconnections.Ifsuchatheoryhadexistedin1917,onewouldhavesaidsimplythatthedifferential1−formsthatonederivescanonicallyfromthemetricandareinstrumentalinobtainingtheRiemanniancurvaturecanbechosenforoneofmanywaysoftransportingvectors.Duetoignorance,twoconceptsbeingrepresentedbythesamesetofcomponentsbecameasetofquantitieswithtwodifferentroles;thissetofquantitieswastreatedasifitwereaconcept.ItwasonlywiththearrivalofCartan’sgeneraltheoryofconnectionsthatitbecamecleartomanyexpertsthatoneisactuallydealingwithtwodifferentconceptsofcurvature,whicharerepresentedingeneralbydifferentdifferentialforms.EverybodyunderstandsnowadaysthattheconnectionneednotbeLC’s.Whatjstillisnotcleartomanyisthatσihasnotceasedtoexistonmanifoldsendowedwithametricbutwithadifferentaffineconnection.Itsimplyhappensthatitonlyplaysitspre-1917role.Correspondingly,wehavetoconsiderEuclideanandmetriccurvatures,respectivelyderivedfromωiandσj.Unfortunately,manyjiexpertsseemtohaveforgottenthattheoriginalroleofRiemann’scurvaturehadtodowiththesolutionoftheproblemofwhetheragivenmetriccanbereducedtotheform(5.1)byacoordinatetransformation,andwithnootherissue.Inviewoftheseconsiderations,readersshouldbynowunderstandthattheEuclideanplanehaszerotorsionandzeroEuclideanandmetriccurvatures.ThepuncturedmetricplaneendowedwiththeColumbusconnectionhaszeroEuclideanandmetriccurvaturesbutnon-zerotorsion.ThesphereendowedwiththeLCconnectionhaszerotorsion,anditsEuclideanandmetriccurvaturesaregivenbythesamedifferentialforms.ThepuncturedsphereendowedwiththeColumbusconnectionhaszeroEuclideancurvature,non-zerometriccurvatureandnon-zerotorsion.Thosecommentsforthespherealsoapplytothetorus,exceptthatitneednotbepunctured.

161Chapter8AFFINECONNECTIONS8.1Liedifferentiation,INVARIANTSandvectorfieldsForachange,readersmayconsiderskippingparagraphsoftheirchoiceinthemiddleofthissectionandthentrytocapturethegistofthelastparagraphs.WewouldnotcareaboutLiedifferentiationinthisbookwereitnotforthefactthat,afterreadingthegreatclassics(CartanandKaehlerinthiscase),onecanonlybepuzzledwhenreadingmodernauthorsinconnectionwiththeconceptsoftangentvectorsandLiedifferentiation.Startinginpage81ofhisextraordinary1922bookonIntegralInvariants[9],Cartandealtwithwhathecalledaninfinitesimaltransformation,nowadaysviewedastheactionofaLieoperator.Heextendedtheactionofsuchoperatorsonordinaryfunctionstodifferentialforms.Adecadelater,thisoperatorwasbaptizedasLiederivative.ItisunderthisnamethatwefinditagaininpapersbyK¨ahler,[46],[48],aswellasSlebodzinski[69](BothacknowledgeCartanforthatextension).Reference[46]isthemostinterestingoneinconnectionwiththepresentdiscussion.CartandidnotdefineaLiederivativeoperatorfordifferentialforms.Hedidnotseeaneedforit.ItisclearinhisworkthattheactionofaninfinitesimaltransformationofafunctiononRn,1∂f2∂fn∂fAf≡ξ+ξ+...+ξ,(1.1)∂x1∂x2∂xnimpliestheformofitsactiononadifferentialformu.Inastrokeofgenius,CartanderivedinafewsimplestepstheformulaAu=(du)A+d[u(A)],(1.2)whichallowsonetoeasilycomputeAusince(du)Aandu(A)arewhatinmod-erntermsiscalledtheevaluationofduanduonavectorfieldAwiththesame143

162144CHAPTER8.AFFINECONNECTIONScomponentsasCartan’sinfinitesimaltransformation.Thisisawellknownthe-oreminthetheoryofLiederivatives,thoughthistermwasnotyetinuse.Notice(a)theabsenceofasymbolforpartialdifferentiation,(b)thatapartfromAandu,thereisin(1.2)onlytheinvariantoperatord(whichmakesclearthatAisacoordinateindependentconcept),and(c)Acomesattheendonbothtermsontheright.Thisdoesnotimpedethevalidityof(1.2)whenAisafunctionofsurfacesandhypersurfaces;theexteriorcalculusisthesameforskew-symmetricmultilinearfunctionsofvectorsasforintegrands(i.e.functionsofhypersurfaces).WithAgivenasi∂A=ξ,(1.3)∂xiK¨ahlerderivestheformula[46]i∂uiAu=ξ+dξ∧eiu.(1.4)∂xiTheoperatoreremovesthefactordxifromu,withsign(+,−,+,−...)depend-iingonthepositionofdxiintheproduct).Onceagain,theactionoftheoperator(1.3)ondifferentialformshasbeenderivedfromitsactiononscalar-valued0−forms.Directproofoftheequivalenceof(1.2)and(1.3)issimple[82].LetusbemorepreciseregardingwhatK¨ahlerdid.Acoordinatetransfor-mationallowedhimtowritethepull-backoftheoperator(1.3)as∂A=,(1.5)∂ynsince∂/∂yi(i=1,...,n−1)arealwayszeroinhisnewcoordinatesystem.Afterperformingthedifferentiation(1.5)ofu,K¨ahlerundoesthepull-back,i.e.returnstoxcoordinates,thusobtaining(1.4).Again,theLiederivativeoffunctionsonRnimplieswhattheLiederivativeofdifferentialformsis.Onemoretreatment,namelybySlebodzinski[69],hasthesameimplication.Inthemodernliteratureondifferentialgeometry,thedominantdefinitionoftangentvectorAatapointofadifferentiablemanifoldis(1.3).OnethengivesthenameofLiederivativetoanewoperatorondifferentialforms,basedontheconceptofflowofavectorfield.Theactionofthisoperatorisnotdifferentfromwhatwehavejustreported,thoughonemaywonderwhyintroducetheconceptinthemodernway,whichlooksadhocandcausesalotofconfusion?(justlookforthetermLiederivativeintheweb).Cartan,K¨ahlerandSlebodzinskiobtained(1.4)inrespectivedifferentwayswithoutresortingtoanewdefinition.Forpracticalpurposes,youcanthenmakeadefinitionofLiederivativeby(1.3)ofadifferentialform,likeK¨ahlerdid.Intheend,itisamatterofextendingtheconceptofpartialderivativetodifferentialforms,startingwiththeelementsofabasisofdifferential1-forms.Assumethatwehadbeengiventheoperator(1.5)ratherthan(1.3)itself.Tobemorerelevant,assumethatynistherotationangleφaroundanaxis

1638.1.LIEDIFFERENTIATION,INVARIANTSANDVECTORFIELDS145ofcylindricalsymmetry.∂u/∂φisthepartialderivativewithrespecttotheparameterofthe1-parametersymmetrygroup.Achangeof∂/∂φtoCartesiancoordinatesyieldsx∂y−y∂x.Thiscorrespondstojustthefirsttermontherightof(1.4).ButthatissobecausethatconceptofpartialderivativeispeculiartofunctionsonRn,nottofunctionsofhypersurfaces.Ifuisgivenintermsofasystemofcoordinates(x)insteadofasystemofcoordinates(y),whereynequalsφ,oneshoulddifferentiatewithrespecttoφnotonlythecoefficientsaofu,butalsothebasis(dxi1∧dxi2∧...).Oneshouldobtainthepartialsi1i2...∂(dxi1∧dxi2∧...)/∂φ,whichiswhatK¨ahlerexplicitlydid.Wereferreaderstothesource[46](ItisinGermanbut,forthisspecificpurpose,readersneedonlyreadtheformulas).Letuslookattheproblemfromtheperspectiveofinvariantoperators.Letynbethespecificcoordinatewithrespecttowhichwewishtodifferentiatewhileeverythingelseremainsconstant.Wewritetheinvariantdasn−1i∂n∂d=dy+dy.(1.6)∂yi∂yni=1ThedifferentialsdyareinvariantssincewehaveByi(B)−yi(A)=dyi,(1.7)Aregardlessofwhetherwepulldyitoanothercoordinatesystemtoperformtheintegration.Westillobtainyi(B)−yi(A)becausewearedealingwithfunctionsonthemanifold.Ifwemanagedtochoosecoordinatessuchthatn−1i∂dy(1.8)∂yii=1wereaninvariant,sowould∂/∂yn,asaconsequenceoftheinvariantcharacterofdandofthedifferencen∂d−dy.(1.9)∂ynK¨ahlerdevisedacoordinatesystemsuchthat(1.8)wouldbezero[46].Whatthemodernliteraturecallsatangentvectorfieldis,forCartanasforK¨ahler,aLieoperatororLiederivative.OneneedsaconceptofvectorfielddifferentfromthemodernoneifonewantstosticktotheletterandspiritofiCartanandK¨ahler.Incidentally,dξ∧eiuisspininthecaseofrotations,asK¨ahlershowed[48].Itisasophisticatedtheoreticaldevelopmentthatinvolves,inaddition,metricstructure(inthiscase,butnotrequiredtosimplyobtain(1.4)).Beasitmay,readersshouldaskthemselveswhatisforrotationsthelasttermof(1.4).CartanandK¨ahlerneverreferredto(1.1)asvectorfield.Forthem,vectorfieldsarepassiveobjects,notdifferentialoperators;theydonotactonanything.Weareinterestedintheirviewbecauseofitssignificanceforrelativisticquantum

164146CHAPTER8.AFFINECONNECTIONSmechanics[48],[83]andpresumablybeyond(Seelastpartofthisbook).Ourproblemthenistodefineatangentvectorasapassiveoperator.Suchaconceptisnotnew,butlargelyoverlooked.Becausedifferentiationsareoftheessenceindealingwithtangentvectors,weshallhavetodealwithparametrizations,while,atthesametime,definingaparameter-independentconcept.Onehastheintuitivenotionofcommontangencyofcurvesatapoint,throughtheirtangencytothesamestraightlinethroughthepoint.Wenowwishtoformulateaconceptofequivalenceofcurvesthatgointhesamedirectionatapointandatthesameratewithrespecttoacommonparameter.Inordertohavethesameparametrizationonallcurves,weuseoneofthecoordinatesasparameter,thesameoneforallcurvesthroughthepoint.Anycoordinateinacoordinatesystemisinprincipleagoodparameterexceptoccasionally.Thus,whendealingwithcirclesontheplane,centeredattheorigin,wecannotuseasparametertheradialcoordinateρ,sinceitdoesnotchangeaswemoveonthecircle.Letxi,forgiveniandcoordinatepatchU,bethechosenparameter.Callity.Letφ1andφ2besmoothfunctions(fromintervalsI1,I2onthereallinetoamanifoldM)suchthatφ1(y)=φ2(y)forthevalueofthecoordinatey.Wesaythatφ1andφ2havethesametangentatPif(d/dy)(fU◦φ1)and(d/dy)(fU◦φ2)takethesamevalueatthatpoint(ReviewthefirstparagraphofChapter2,section8).Usingtheconceptofdifferentiablemanifoldoneshowsthatifthisconditionissatisfiedinonecoordinatepatch,italsoholdsinanyothercoordinatepatch.Readerswhomayhavehadproblemswiththepreviousparagraphneedsim-plythinkofthetangenttoacurveatapointasasetofquantities∂xi/∂λ,whereλistheparameteronthecurve.Thedigressioninapreviousparagraphaboutthechoiceofacoordinateasthecommonparameterdealswiththeneedtocom-pareratesofchangeondifferentcurves,andalsoondifferentparametrizationsofthesamecurve.OnenowdefinesatangentvectoratPasanequivalenceofcurvesthathavetherethesametangent.Thisiscloselyrelatedtotheintuitiveideaofidentifyingsmallpiecesofthemanifoldwithsmallpiecesofaffinespace,exceptthatwenowdoitthroughcurves.AsmallpieceofcurvebetweenitspointsP(y)iandP(y+Δy)iscloselyrepresentedbyasetofquantitiesdxΔy.Thereis,dyPtherefore,anobviousandthusnaturalphysicalinterpretationoftangentvectorsassuchequivalences(readequivalenceclasses).Wenextendowtangentvectorsatanygivenpointwithastructureofvectorspaceandbuildbasesthereof.And,usingthegroupGL(n),wecreateafiberateachpointofthemanifold.WehavethusgotaframebundleverymuchliketheframebundleFn.Thedifferenceis,however,thatwedonothaveprojectionmapslikewehadinaffinespacebecausewecannotassignvectorstoarbitrarypairsofpointsofthebasemanifold.Correspondingly,weshallnothave,ingeneral,rectilinearcoordinatesAi.Tosummarize:onanarbitraryn−dimensionaldifferentiablemanifoldwe

1658.2.AFFINECONNECTIONSANDEQUATIONSOFSTRUCTURE147introduceabundleoftangentvectorspaces,i.e.atangentvectorspaceateachpointofthemanifold.Pictorially,wedividethelatterintosmalloverlappingpieces,whichwematchwithsmallpiecesofaffinespacesthroughthetangentvectorspaces,thepointoftangencyplayingtheroleofzero.8.2AffineconnectionsandequationsofstructureAmoreformalapproachtothesubjectofthissectionwillbegiveninthelastoneofthischapter.Weshallexplicitlyindicatewhetherasubscriptorasuperscriptcomesfirstinordertolaterdealwithissuesofskew-symmetry.Bywayofthebundle,wegiveinsimpletermstheintegratedconceptsofconnectionsandequationsofstructure,formallyintroducedattheendofthechapter.AdifferentiablemanifoldMofdimensionnissaidtobeaffinelyconnectedifweassociatewithitanothermanifoldendowedwithn+n2linearlyindependentdifferentialformsωiandωjwithpropertiesthatguaranteethatthesystemofk“connectionequations”ijdP=ωei,dei=ωiej(2.1)willmakesmallpiecesofMlooklikesmallpiecesofaffinespaceofthesamedimension.Inthetableattheendofthechapterweusenotationthatreflectsthatdin(2.1)doesnotrepresentdifferentiationsofanything,asexplainedinitsfootnote.TheωimustbelinearlyindependentcombinationsofthedifferentialofthecoordinatesonMandonlythey.Inotherwords,theymustbehorizontal.Thecoefficientsinthelinearcombinationwilldependnotonlyonthex’sbutalsoonthen2additionalcoordinatesdefiningthefibers.Asaconsequence,thedωijwillnotbehorizontal.Theω,thoughnothorizontal,mustbesuchthat,whenijpulledtothefibers,theybecometheωiofthelineargroup.“Pullingtothefibers”isthetechnicalwayofsayingthatwesetthedxi,equivalentlytheωi,tozero.Wefurtherrequirethat,whenwecomputedωi−ωj∧ωianddωj−ωk∧ωj,jiikweobtainexpressionswhicharequadraticexpressionsintheωjexclusively.WemaythenwriteΩi=dωi−ωj∧ωi,Ωj=dωj−ωk∧ωj,(2.2)jiiikwhereiikl1iklΩ≡Rkl(ω∧ω)=Rklω∧ω,(2.3a)2jjkl1jklΩi≡Rikl(ω∧ω)=Riklω∧ω.(2.3b)2

166148CHAPTER8.AFFINECONNECTIONSNoticethatdωianddωjmustbenon-horizontal,sothattheirnon-horizontalicontributionscanceloutwiththenon-horizontalcontributionsofthelasttermsinEqs.(2.2).Forsimplicity,wetakeeitherk

1678.2.AFFINECONNECTIONSANDEQUATIONSOFSTRUCTURE149Weformallydifferentiatethesystem(2.1)toobtaind(dP)=(dωi−ωj∧ωi)e(2.8a)jijkjd(dei)=(dωi−ωi∧ωk)ej.(2.8b)Thelefthandsidesof(2.2)areshorthandexpressionsfortherighthandsides.Equations(2.5)cannowbegiventhecompactformid(dP)=Ωei(2.9a)jd(de)i=Ωiej.(2.9b)Oncethisisunderstood,onemayusethetermtorsiontorefertoanyoftheleftandrighthandsidesof(2.8a)and(2.9a).Similarly,onemayusethetermaffinecurvaturetorefertoanyoftheleftandrighthandsidesof(2.8b)and(2.9b).Weproceedtorepeattheformulasofsection5.6forthecomponentsofthecurvature,sincetheywillguideuswhenmakingsimilarimportantconsidera-tionsforthelesswellunderstoodconceptoftorsion.Wemomentarilyintroduceasmallchangeofnotationalconventiontomakethepointthat,exceptionally,therewillnotbedualitybetweenthedifferentbasesoftangenttensors,ontheonehand,andbasesofdifferentialforms,ontheother.Onsections,weexpressωjasΓjdxkratherthanasΓjωkinordertoiikiksimplifydifferentiations,sinceddxkiszerobutdωkisnot.WeprimedthejindexkratherthanthemainlinecharacterΓtomakeexplicitthatmakesireferencetoa,ingeneral,non-holonomicfieldofvectorbases,andkbelongstoaholonomicorcoordinatebasisofdifferential1−forms.Wethenhave:dωj=d(Γjk)=(Γjjl∧dxk),(2.10a)iikdxik,l−Γil,k)(dx−ωm∧ωj=(Γmj−Γmj)(dxl∧dxk),(2.10b)imikΓmlilΓmkΩj=(Γjjmjmjl∧dxk).(2.10c)iik,l−Γil,k+ΓikΓml−ΓilΓmk)(dxjDefiningRasilkj1jjmjmjRilk≡(Γik,l−Γil,k+ΓikΓml−ΓilΓmk),(2.11)2wecanwriteΩi=Ril∧dxm.(2.12)jjlmdxItisalsocommontodefineRj=Γjjmjmjilkik,l−Γil,k+ΓikΓml−ΓilΓmk.(2.13)Theni1ilmΩj=Rjlmdx∧dx.(2.14)2Atthispoint,itisadvisabletoreturntoanexpansionintermsofωl∧ωm,sothattherewillbecorrespondence(i.e.duality)betweenbasesinthealgebras

168150CHAPTER8.AFFINECONNECTIONSofdifferentialformsandofvaluedness.Substitutionofthedxiintermsoftheωjin1ilm1ilmRjlmdx∧dx=Rjlmω∧ω(2.15)22allowsonetorelatecoefficients.Herethechangeofthecomponentsofthecurvatureisbilinear(concernsonlytheindiceslandm)ratherthanquadri-linear.Thereasonisthat,wheninordertosimplifydifferentiationsweexpressωjof(2.9)asΓjdxkratherthanasΓjωk,wechangebasesofdifferentialiikikformsbutnotbasesoftangentvectorsandtensors.Fromthispointon,thebasesofvaluednessandthebasesofdifferentialformswillcorrespondtoeachother.Considernowthetorsion.Intermsofcoordinatebasisfields,Ωiequals−dxj∧ωi.TheequationjiiiRjk=Γkj−Γjk(2.16)follows.Allindicescorrespondheretoholonomicbases.Thishasgivenrisetothestatementthatthetorsionistheantisymmetric(whatwehavecalledskew-symmetric)partoftheconnection.Thatisagenerallyincorrectstatementsinceitdoesnotconcernarbitrarybasesofdifferentialformsandframefields.Whentheyarearbitrary,onehasRi(ωj∧ωk)=(Mi+Γi−Γi)(ωj∧ωk),(2.17)jkjkkjjkwhereMi(ωj∧ωk)standsfordωi.Equation(2.16)doesnotfollowingeneral.jkNoticethedifferentpositioningoftheindicesjandkinRrelativetoΓ.Theskew-symmetryofRiin(2.16),inadditiontopertainingtocoordinatejkbasisfields,concernsthefirstandthirdindicesinΓ.Ontheotherhand,theskew-symmetryωij+ωji=0discussedinchapter6(Eq.(1.5))pertainstoorthonormalbasisfieldsandconcernsthefirstandsecondindicesinthegammas.OnlyonEuclideanspaceswehaveframefieldswheretheskew-symmetry0=Γkij−Γjik(zerotorsion)andthesymmetryΓijk+Γikj=0(metriccompatibility)coexist.IngeneralizationsofEuclideanspacesknownasRiemannianspaces,bothequationscanbethecase,butnotatthesametime.Thefirstonerequiresholonomicbases.WemustthenuseEq.(2.11)ofthatchapter,llgij,m−Γimglj−Γjmgli=0,(2.18)whichimpliesΓijk+Γikj=gij,k(2.19)ratherthanΓijk+Γikj=0.Inlaterchapters,readerswillbeinabetterpositiontounderstandallthis.8.3TensorialityissuesandseconddifferentiationsInthissection,wepreparethegroundforthestudyofseconddifferentiations.Wefirstconsidertensorialityissuesrelatedtod(dej).Unlikedej,ittransforms

1698.3.TENSORIALITYISSUESANDSECONDDIFFERENTIATIONS151linearlyunderachangeofframefield.Itdoessoinsuchawaythatitscontrac-tionwiththecomponentsvjofavectorfieldisinvariant.Onesometimesfindsintheliteraturethestatementthatatensortransformsinsuchandsuchwayunderacoordinatetransformation.Tensorsdonottrans-form.Theyareinvariants.Theircomponentsdo,underchangesofsectionoftheframebundle.ThechangesofsectionareelementsofthegroupG0,differentelementsingeneralatdifferentpointsineachsection.G0actsp+q+rtimesinthechangeofbasisof(p,q)−valueddifferentialr−forms,unlesswechangeonlythebasisofvaluednessoronlythebasisofdifferentialforms.When,insomeofjtheliterature,scalarfunctionsγiaresaidtobetensor-valued,authorshaveinjiimindγiφej.Theyfailtoexhibitthebasisφejfor(1,1)-tensorvaluedness.Inanequationsuchas(5.1)ofchapter3forthetransformationofcompo-nentsofavectorfield,onlycoordinatesonthebasemanifoldshowup.ThisdoesnotconflictwithwhatwehavejustsaidabouttransformationswhicharemembersofG0,whichthusdependonthecoordinatesinthefibers.Atanyjgivenpoint,theparametersAioftheelementG0isthematrixwhosecompo-nentsarethe∂xj/∂xievaluatedatthatpoint.Theybringupthecoordinatesinthesectionbypullback.Wesawintheprevioussection—thoughitmusthavebeenobviousevenbefore—thatthetensorialtransformationofcomponentsonpull-backstosec-tionsreflectshorizontalityinthebundle.Buthorizontalityconcernsthescalar-valueddifferentialformthatmultipliesatensortoformatensor-valueddiffer-entialform.Thequestionis:howdoesonereconcilethefactthat,whereastensorialityinvolvesallindicesinthecomponents,horizontalityonlyinvolvesthedifferentialformindicesamongthem?jijiAninvariantofthetypeγiφejisnotsodueto“itslooks”,sinceωiφejhasjthesamelooksbutisnotinvariantfromsectiontosection.Assumethattheγijjandωiweredifferential1−forms.Inthebundle,wherewecannotsettheAijtobefunctionsofx,itistheabsenceofthedifferentialsoftheAithatdefinesjhorizontality.Thedifferencebetweenthetensorialtransformationofγandtheijnon-tensorialoneofωiliesexclusivelyinthatthefirstoneishorizontalandthesecondoneisnot.Ithasnotingtodowiththeiandjindices.Thelineartransformationpropertiesforvaluednessindicesisguaranteedbyconstruction,jjnamelybytheactionofthegrouponthefibersforbothγiandωi.Thatiswhyhorizontalityguaranteestensoriality,i.e.lineartransformationinconnectionwithallexplicit(i.e.valuedness)andimplicit(i.e.differentialform)indices.Leteandebevectorbasesattheintersectionoftwogivensectionswithiithefiberatx.Wehavejei=Ai(x)ej.(3.1)Formaldifferentiationyieldsjjdei=dAiej+Aidej.(3.2)Thefirsttermontherighthandsideof(3.2)isnotingeneralalinearcombina-tionofthedej.Sincethesecondtermis,theirsumand,therefore,thelefthand

170152CHAPTER8.AFFINECONNECTIONSsideoftheequationarenot.Thusthedei’sdonottransformlinearly.But,ifwedifferentiate(3.2),wegetjd(dei)=Aid(dej),(3.3)sincethefirstofthefourtermsresultingfromdifferentiatingtherighthandsideof(3.2)iszeroandthenexttwocanceleachotherout.Thisequationshowsthatd(dei)transformsinthefiberslikethebasisitself,sothatcontractionwiththecomponentsviisthesameinallsections.Onceagain,thisisnotthecasewithvjde.ComparisonofEqs.(3.2)and(3.3)exhibitsthecontrastbetweenjtherespectivenon-tensorialandtensorialcharacterofthesubscriptiinthetwocases.Whenonedefinesdei—whichiswhatwedidintheprevioussection—westillhave,asinaffine-Kleingeometry,dv=v(P+dP)−v(P)=vi(P+dP)e(P+dP)−vi(P)e(P)ii=dvie+vide.(3.4)iiWiththenotationP+dPwemeanthat,whenevaluatingdv,thecoordinatefunctionsarexi+dxi.Alltheformulasofchapter5applyhere.Wehaveinparticularthatiiiidvei+vdei=dvei+vdei.(3.5)Butwedidnothaveequalitiesdvie=dvieandvide=videthen,andweiiiidonothavethemnow.Insearchfordeepunderstanding,weproceedtoformallydifferentiatetwiceithefieldofbasesofvector-valued1−forms,φej,soastothendifferentiatejii(1,1)-valueddifferential0−forms,γiφej.Asafirststep,wefindd(dφ):d(dφi)=d(−ωiφj)=−(dωi)φj+ωi∧dφk=−Ωiφj.(3.6)jjkjiWhenwedifferentiateφejtwice,twoofthetermsresultingfromtheseconddifferentiationcanceleachotheroutandweobtainiiid[d(φej)]=[d(dφ)]ej+φddej.(3.7)Therefore,using(2.10b)and(3.6)in(3.7),wefurthergetd[d(φie)]=−Ωiφke+Ωkφie.(3.8)jkjjkTheseconddifferentialofγiφjeisnowobvious.Weabbreviateγiasγandjijjφeiasf.Thus:d(γf)=(dγ)f+γdf.(3.9)Furtherdifferentiationyieldsdd(γf)=−dγ∧df+dγ∧df+γd[d(f)],(3.10)

1718.4.DEVELOPMENTSANDANNULMENTOFCONNECTION153wherewehaveusedddγ=0.Usingnow(3.8),wefinallygetdd[γj(φie)]=γj(Ωkφie−Ωiφke).(3.11)ijijkkjWeleaveitforinterestedreaderstheminormodificationsrequiredinthelastjthreeequationswhentheγi’sarenot0−forms.Nonewtermsappear,butchangesinsignmayemergeintheprocessofexteriordifferentiation.8.4TangentdevelopmentsandannulmentofconnectionatapointTheconnectionequationforeicanbewrittenasjei(P+dP)=ei(P)+ωi(P)ej(P),(4.1)tobeintegratedontinycurves.AsmallvectorΔPisthusassignedtotheendpointsofthecurve.Wereversetherolesoftheinitialandendpointsofthecurveanddenoteeiasa.Wegetja(P)=ei(P+dP)−ωi(P+dP)ej(P+dP).(4.2)Theminussignreflectschangeoforientationwhenevaluating(4.1)(i.e.inte-jgratingωi)onparametrizedtinycurves.Wethusrewrite(4.2)asa(P)=aj(P+ΔP)e(P+ΔP),(4.3)jwhereai=1−ωi,(nosum)(4.4a)ijja=−ωi,j=i,(4.4b)jjωexceptionallyrepresentingheretheactualevaluationofωinthetinycurve.iiWehavethusexpressedthevectoraatPintermsofabasisatP+dP.Weshallnowremovetherestrictionofthecurvebeingtiny.Onparametriza-tionsofcurves,allpartialderivativesbecomederivativeswithrespecttotheparameter,λ.Wetakeabasis(ei)atP(λ=0)as“initialcondition”fortheintegrationalongacurveofthesystemofdifferentialequationsjdei(λ)=ωi(λ)ej(λ).(4.5)Forclarity,weshalldenotetheseinitialvaluesof(ei)as(ai).Theresultofintegrating(4.5)isjei(λ)=Ci(λ)aj.(4.6)jIfwedenotethematrix[Ci]asCandsolveforaj,weget,withsomeabuseofnotation,{a}={e(0)}=C−1{e(λ)}.(4.7)jj

172154CHAPTER8.AFFINECONNECTIONSEquation(4.7)showshowthebasisatP(0)isexpressedintermsofthebasisatP(λ).ThematrixC−1dependsonconnectionandcurvebetweenpointsP(0)andP(λ).Thedependenceoncurvemeansthatwedonothaveherearelationofequivalence;thetransitivepropertyisnotsatisfied.Weshallsaythatavectorvi(λ)e(λ)atP(λ)istheparalleltransportedivectorviaalongacurveγfromP(0)toP(λ)ifiiiv(λ)ei(λ)=vai.(4.8)Thecomponentsvi(λ)canbeobtainedbysubstitutionof(4.7)in(4.8)andequatingcoefficientsonbothsides.vi(λ)e(λ)isaconstantvectorfieldontheicurve.jWenowexplicitlyshowthatωicanbemadetovanishatanygivenpointbyappropriatechoiceofbasisfield.Werewrite(4.1)asjei(x+dx)=ei(x)+ωi(x)ej(x).(4.9)Ontheotherhand,rewritingxas(x+dx)−dx,wehavejei(x)=ei(x+dx)−ωi(x+dx)ej(x+dx).(4.10)Fromtheseequations,wegetjjωi(x)ej(x)=ωi(x+dx)ej(x+dx),(4.11)and,therefore,(4.9)canfurtherbewrittenasjei(x+dx)−ωi(x+dx)ej(x+dx)=ej(x).(4.12)Definingtheleftandrighthandsidesof(4.12)respectivelyase(x+dx)andje(x),(4.12)impliesje(x+dx)=e(x),(4.13)jjor,inotherwords,jωi(x)=0.(4.14)8.5InterpretationoftheaffinecurvatureWeproceedtogiveageometricinterpretationofthecurvatureandofd(dei).LetP,M,MandPbethefourcornersofasmallcurvilinearquadrilateralonthemanifold(Figure6).Sincetheresulttobeobtainedistensorial,itwillbevalidinanysection.Figure6hasamnemonicpurpose.Itisnotadevelopmentinthetangentplane,whichdoesnotcloseingeneral.Nordoesitbelongtothemanifoldsinced1Pandd2P,belongtotangentplanes(Somewouldfindmoreappropriatetowrited1Pandd1Pasiftosignifythatwearenotdealingwithchangesofavectorbutwithvectorassociatedwithpairsofpointsinthemanifold).Wehave

1738.5.INTERPRETATIONOFTHEAFFINECURVATURE155closedtherectangleinordertomakeclearthatwearegettingtothesamepointofthemanifold,P,intwodifferentways.Wewritetheparallel-transportedei(P)fromPtoMas(ei)P→M.Theequationsthatfollowarethenobvious.(ei)P→M=ei(P)+d1ei(P),(5.1)(ei)P→M→P=(ei)P→M+d2[(ei)P→M],(5.2)(ei)P→M→P=ei(P)+d1ei(P)+d2ei(P)+d2d1ei(P).(5.3)whered1eiandd2eiarenotdifferentialsbutevaluationsontwotinycurves.P’dP1M’dP2dP2Md1PPFigure6:Assignmentofsegments,notdevelopmentinEuclideanspaceParalleltransportalongtheotherpath,P→M→P,yields(ei)P→M→P=ei(P)+d1ei(P)+d2ei(P)+d1d2ei(P),(5.4)Wesubtract(5.4)from(5.3)andobtain(ei)P→M→P−(ei)P→M→P=d2d1ei(P)−d1d2ei(P).(5.5)Theexpansionofdde,whichbelongstothepathP→M→P,is21ijkjd2d1ei=d2[ωi(1)ej]=dx(1)d2(Γikej)(5.6)=dxk(1)dxl(2)Γje+dxk(1)dxl(2)ΓmΓje.ik,ljikmljClearlythesymbolsdxk(1)anddxk(2)areincrementsinthecoordinates.Fortheotherpath,weinterchangethenumbers1and2in(5.6),exchangethedummyindiceskandlandgetdde=dxk(1)dxl(2)Γje+dxk(1)dxl(2)ΓmΓje,(5.7)12iil,kjilmkj

174156CHAPTER8.AFFINECONNECTIONSWesubtract(5.7)from(5.6)and,tofacilitatelaterdevelopments,wewritedxk(1)anddxk(2)asdxk(g)anddxk(n)respectively.Wethusgetdde−dde=dxk(g)dxl(n)[Γj−Γj+ΓmΓj−ΓmΓj]e.(5.8)21i12iik,lil,kikmlilmkjThecontentsofthesquarebracketisskew-symmetricinkandl.Iftheproductdxk(g)dxl(n)weresymmetric,therighthandsideof(5.8)wouldvanish.Butitisnot.Thus,forinstance,dx1(g)dx2(n)isnotthesameasdx2(g)dx1(n).Recallthatdx,dy,...arenotsmallincrementsbutfunctionsofcurves.Ontheotherhand,dxk(g)anddxl(n)representtheresultofevaluations(readintegrations)ofdxkanddxlonsmallcurves.Readersmaythinkofdxk(g)anddxl(n)as,forexample,ΔxandΔy.AndtheymayprefertousethenotationΔ2Δ1eiinsteadofd2d1ei.Alldependsonwhatonewishestoemphasize.Wereplacesmallquantitieswithdifferentialsforobtainingadifferentialequation.Wethusstatetheresultofouranalysissofarasj(ei)P−M−P−(ei)P−M−P→ddei=Ωiej.(5.9)Noticetheorderingonthelefthandsideof(5.9).PuttingtogetherP−M−Pand,inreverseorder,P−M−P,wegetP−M−P−M−P,whichamountstoreturningtothesamepointfollowingaclosedcurvewithanti-clockwiseorientation.Equation(5.9)yieldsthegeometricinterpretationoftheaffinecurvature,whichcanalsobereadfromddv−ddv→vid(de)=viΩje=ddv.(5.10)2112iijThedifferencebetweenavectorandthesamevectorafterwetakeitalongaclosedcurveisobtainedbyintegrationofviΩjeonthesmallsurfaceenclosedijbyatinyclosedcurveonthemanifold.Thefinalstepintheinterpretationofthecurvaturetakesusbacktosection7ofchapter5,whereoneconnectsEq.(7.7),whichisanotherwayofwriting(5.10),withdefinedasji≡Ωiφej.(5.11)(Welackaboldfacedsymbol).8.6ThecurvaturetensorfieldassociatedwiththecurvaturedifferentialformWhenwetransportavectorfromonepointtoanearbypointalongtwodifferentpaths,wegetingeneraltwodifferentvectorsattheendpoint.Theaffinecurvatureallowsustoexpedientlycomputethedifferencebetweenthetwo.Forafutureapplication,weshallspecializethisproblemtowhenthetwopathsmakeacurvilinearquadrilateralconstitutedbytheintersectionofcurvesoftwodifferentcontinuousfamiliesofcurves.Wecanapproximatetheintegralsby

1758.6.THECURVATURETENSORFIELD157M’g(u)2Mn()n()21nPtg(u)1P’Figure7:Notationforpairsofintersectingcurvesreplacingthesidesofsmallcurvilinearquadrilateralswithsegmentstangenttothecurvesinvolved.Indoingso,thecomponentofthetensor-valueddifferential2−formcurvaturebecomethecomponentsofa(1,3)-tensor.InFigure7,wedenoteasgandnthemembersofthetwointersectingfamiliesofcurves.Inthenextsection,oneofthefamilieswillbemadeofautoparallels(linesofconstantdirection).Insection3ofchapter10,theywillbegeodesiccurves(i.e.curvesofstationarydistance).Wemoveavectorfromonepointtoanotheralongagcurveandcontinuebymovingitonanncurvetoa“finalpoint”.Wealsomovebetweenthesamepoints,firstonanncurveandthenonagcurve(Noticetheslightchangeofthelabellingofthepointsinfigure7relativetowhatitwasinfigure6).Forsimplicityweusecoordinatebases.EvaluationsonthecurvesofFig.7undertheassumptionthattheyareverysmallyieldsddv−ddv=viRjdxk(g)dxl(n)e,(6.1)2112ikljwheredxk(g)anddxl(n)areincrements,notdifferentials.Noticethat,atthispoint,wearenotdealingwiththecurvaturedifferentialform,sincewehavealreadyperformedits“integration”onthesurfacedelimitedjbythesmallpiecesofcurves.TheRarethecomponentsofa(1,3)tensorfieldiklφiRjφkφlethathasjustbeenevaluatedonthetripleofvectors(v,dxk(g)e,ikljkdxl(n)e)tangenttothemanifoldatP.lThedxk(g)anddxl(n)aretmΔuandnmΔλonthecurvesgandnrespectively.Thetangentvectorstandnareindicatedinthefigure.Wethushaveddv−ddv=viRjtknlΔuΔλe.(6.2)uλλuikljEquation(6.2)lendsitselftoberewrittenasdudλv−dλduvijkllim=vRikltnej,(6.3)Δu,Δλ→0ΔuΔλjjiAgain,theRhavetobeviewedasthecomponentsofa(1,3)-tensorRφ⊗ikliklklej⊗φ⊗φ.Therighthandsideof(6.3)representstheevaluationofthistensoronthetripleofvectors(v,t,n).

176158CHAPTER8.AFFINECONNECTIONS8.7AutoparallelsAnautoparallelisacurvewherethetangentvectordP/dλdefinedbyagivenparametrizationremainsequaltoitselfuptoafactor,whichmayvaryfrompointtopointandfromoneparametrizationtoanother.Foragivenautoparallelcurveandparameteru,wedefineα(u)bydt≡α(u)tdu.(7.1)Wereservethesymboluforparametrizationsyieldingdt=0.(7.2)SincedPonthecurvecanbewrittenastdu,butalsoastdu,wehavetdu=tdu.(7.3)Equation(7.3)impliesthatβ(u)t=t,(7.4)whereβ=du/du.Differentiating(7.4)andtakingintoaccount(7.1)-(7.3),oneobtainsdβt+0=αtdu=αβdut.(7.5)Afirstintegrationyieldsαduduβ=C1e=,(7.6)duand,integratingagain,u=Ceαdudu+C,(7.7)12whichshowshowtoobtainaffineparameters(meaningthoseforwhichdt/du=0)fromnon-affineones.Itisclearthatallaffineparametersaregivenintermsofoneofthembyu=cu+c,(7.8)12asfollowsfrom(7.7)withαequaltozero.Weshallchangeparametersasneededsothatautoparallelssatisfydt=0.Letvbethetangentvectorttoafamilyofautoparallels.Wethenhavedudλtijkllim=tRikltnej,(7.9)du,dλ→0dudλhavingsetdut=0becausedt=0.Wethenhave∂2t=tkRjtinle.(7.10)kilj∂u∂λInalaterapplicationofthisequation,weshallgetabetterfeelingforexpressionssuchastheoneonthelefthandsideof(7.10).

1778.8.BIANCHIIDENTITIES1598.8BianchiidentitiesExteriordifferentiationoftheequationsofstructureyieldsdΩi=−dωj∧ωi+ωj∧dωi,(8.1a)jdΩj=−dωk∧ωj+ωk∧dωj.(8.1b)iikikThedifferentialsoftheω’sareobtainedfromEqs.(2.3)andreplacedin(8.1)toyielddΩi=−Ωj∧ωi+ωj∧Ωi,(8.2a)jjdΩj=−Ωk∧ωj+ωk∧Ωj.(8.2b)iikikIfyoudidsomealgebratogetfrom(8.1)to(8.2),youmadeanunnecessaryeffort.JustreplacedωjanddωkwithΩjandΩiinEqs.(8.1).Thisismoreikthanjustamnemonicrule:wecanignorealltermsthatdonotinvolvetorsionandcurvaturesincetheycanceloutamongthemselveswhenbotharezero.ThisissobecausetheBianchiidentitiesofaffinespaceread0=0.Equations(8.2)areknownasBianchiidentities.Moreoftenthannot,onedealswiththeirpull-backtosectionsoftheframebundle,wheretheyreceivethesamename.Theyareintegrabilityconditionsforthesystemofequationsofstructure,sothatinvariantconnectionformsonthemanifoldexist.Cartanrefersto(8.2a)and(8.2b)asthelawsofconservationoftorsionandcurvaturerespectively.Recallfromthelastsectionofchapter4howhedealtwithconservation.Thestartingpointisadifferentialformwhoseexteriorderivativeiszero.Differentdaughterconservationlawsthenfollowdependingongraderofthedifferentialform,typeofr-surfacethatonechoosesasdomainofintegration,dimensionofthemanifoldandsignatureofthemetric.Thequestionthenis:ifdα=0for,say,avector-valueddifferentialformα,whatisacorrespondingstatementofconservation?Ifthereisageneralruleforsuchdifferentialforms,thefirstBianchiidentity,(8.2a),shouldbeaparticularcaseofconservationofvector-valueddifferentialforms.WeintroducethecompactnotationΩ≡Ωie,≡Ωjφie.(8.3)iijTheBianchiidentitiescanthenbewrittenasdΩ=ωj∧Ωie,d≡0.(8.4)jiTheseareequationsratherthanidentitiesifΩandareexplicitlygivendif-ferentialforms.Andtheyareidentitiesifweinsteadreplacethemwiththeirdefinitionsintermsofthefundamentalinvariantsofthedifferentiablemanifoldandtheirderivatives.

178160CHAPTER8.AFFINECONNECTIONS8.9IntegrabilityandinterpretationofthetorsionTorsionisavector-valueddifferential2−formofcomponents:j1ijkijkΩ≡Rkjω∧ω=Rkj(ω∧ω).(9.1)2Ωieisthevector-valueddifferentialformd(dP),i.e.d(ωie).RememberthatiidPisnottheresultofdifferentiatingsomevectorvaluedfunctioningeneral;bothdPanddeiareeithergivenexplicitlyorasolutionofsomesystemofdifferentialequations,say,theequationsofstructure.Theoptionusedinaffinejspaceofdifferentiatingei=Ai(x)ajandreplacingajintermsofejinordertojobtaindei=ωiejisnotavailableongeneraldifferentiablemanifolds;onesuchexpressionforeiintermsofajatsomepointdoesnotexistunlessthemanifoldisendowedwithequalityofvectorsatadistance,conditioncalledteleparallelism(TP).TheaffinecurvaturewouldhavetobezerooverregionsofMforthatoptiontoapply.Recallthataffinecurvaturecanbeviewedasbeingabouttransportingavectoraroundaclosedcurveandreturningtothepointofdeparturewithadifferentvector.Torsion,ontheotherhand,isaboutrepresentinginaffinespacesmallcurvesofgeneralaffinelyconnecteddifferentiablemanifolds.Thatthepathshavetobeverysmallforintegrationofvector-valuedintegralstomeananythingatallisseldomifeverconsideredinbooksondifferentialgeometry,moreinterestedinformalaspectsthanongeometricinterpretations.Ortheyhaveitwrongforfailingtorefertoverysmallpaths.ElieCartan,authorofthetheoryofaffineconnections,makesemphasistime´andagainonsmallnessofcurvesiftheaffinecurvatureisnotzero.Theresultsobtaineddependonwhatpathischosentobringtothesamevectorspacethecontributionsbysmallpiecesofsurface,contributionswhosesumconstitutes(inthelimit)thevector-valuedsurfaceintegral.Recallthatthose“tinyterms”liveindifferenttangentspaces.Becauseofitsrelevance,wequotefromaseminalpaperbyCartanof1923[11]wherehepresentshistheoryofaffineconnections.Hewrote(minorchangestomoremodernnotationhavebeenintroduced):“Onefinallyarrives,foraninfinitelysmallcontour,totheformuladP=(dωi−ωj∧ωi)e,(9.2)jiidenticaltotheformulafoundforaproperaffinespace.Thevectorseiontherighthandsidearehererelativetoanypoint,providedthatitbeinfinitelyclosetothecontour...”(emphasisinoriginal).Theverylaststatementisobviouslymakingreferencetotheneedtobeusingjustonevectorspace.Ifandwhenthatisthecase,thevectorbasiscanbetakenoutoftheintegral,whichjustifiestheuseofStokestheoremwhendealingwithvector-valueddifferentialforms.Equation(9.2)assignsavectortoapath,vectorwhichiszeroinaffinespacebutnotingeneral.InthecaseofPPMMofFigure7,itsrepresentation

1798.10.TENSOR-VALUEDNESSANDTHECONSERVATIONLAW161onaffinespaceisasetoffourvectors,oneafteranother,tomakea“quasi-quadrilateral”.Theintegralthenrepresentsthetinyvectorjoiningthetwoendsofthequasiquadrilateral.Cartanmakesthesameconsiderationaboutinfinitelysmallcontourswhen,inthesamesection,hereachesanequationthatparallels(9.2)butinvolvesthecurvature.Finally,wheninthesamespiritCartanexplainsthegeometricsignificanceoftheBianchiidentities,hewrites:“Thegeometricsumoftheinfinitelysmalldisplacementsassociatedwiththedifferentelementsofaclosedsurfaceiszerowhentheclosedsurfaceisinfinitelysmall.”([11],fulltestinitalicsinorigi-nal,butboldfacehasbeenadded).NoticeCartan’sinsistenceonsmallnessofintegrationdomains.Tosummarize,affinetorsionisaboutrepresentingonaffinespaceitself(i.e.onthecorrespondingKleinspace)tinyclosedcurvesofanaffinelyconnectedmanifoldand“quantifying”itsfailuretocloseintheformofavector,whichoneobtainsthroughintegrationofthetorsion.Letitnotbeforgottenthattherepresentationofclosedcurvesfailstocloseevenifthetorsioniszerobuttheaffinecurvatureisnot.Butthereasonforthefailuretocloseisnotthesameonewhenthetorsioniszeroaswhentheaffinecurvatureiszero.Ingeneral,bothtorsionandcurvaturewillcontributetothefailuretoclose.8.10Tensor-valuednessandtheconservationlawInthissectionwespeakofthelimitationsoftheconservationlawindealingwithtensor-valueddifferentialformsifthecurvatureoftheaffineconnectiondoesnotvanish.Intensorcalculus,thecovariantderivativeistheallpervadingderivative.Incalculusofdifferentialforms,itemergesonlyinthedifferentiationoftensor-valueddifferential0−formswithwhatCartanandK¨ahlercallexteriorderiva-tive.Inrecognitionofthepracticeinmostoftheliterature(notnecessaryhere)ofthetermexteriorcovariant,weshalltaketheintermediatecourseofwritingdown“exterior(-covariant)”whereCartanandK¨ahlerwouldusethetermex-terior.Thenatureoftheobjectbeingdifferentiateddetermineswhetheronegetswhatgoesintheliteraturebythenamesofexterior,covariantorexteriorcovariantderivatives.Theconservationlawofvectorfieldsisofparticularinterestbecauseofitsrelativesimplicity.Itisthenextstepupinvaluednessafterscalarvaluedness.Inchapter4,section7,welearnedthat,ifαisscalar-valuedanddα=0insomesimplyconnecteddomainR,wehaveα=0.(10.1)∂RIf,inaddition,αisa0−form,f,wehaveforanytwopointsinR:f(B)−f(A)=0.(10.2)

180162CHAPTER8.AFFINECONNECTIONSLetusnowreplacescalar-valuednesswithvector-valuedness.Theexteriorderivativeofavectorfield(inCartanandK¨ahler’sterminology)isavector-valueddifferential1−form,iijdv=d(vei)=v;jωei,(10.3)wherevi≡vi+vkΓi,(10.4);j/jkjwhichisknownintheliteratureasthecovariantderivative.Inasimilarwaytotheconservationofα,onewouldbetemptedtosay(temptationthatwemustresist)that,ifdv=0,Stokestheorem,dv=v,holdit!(10.5)R∂Ryieldsv=0,holdit!(10.6)∂Rwhere∂RisthepairofendpointsAandBofthecurveR.WewouldthenconcludethatvB=vAinsimplyconnecteddomains.Thatiswrong.vBandvAbelongtotwodifferentspacesanditsequalityisnotevendefinedingeneral.Thiswrongargumentemphasizessomethingaboutintegrationofanythingthatistensor-valued,andvectorvaluedinparticular:ingeneral,thoseintegrationsarenotdefined.Thoseintegrationsare,however,definedifthereareconstantbasisfieldsinourmanifold.SuchisthecaseinaffineandEuclideanspaces,andintheirgen-eralizationsendowedwithteleparallelconnections.Usingconstantbasisfields,therelationbetweenvector-valuedquantitesbecomesarelationbetweentheircomponentswithrespecttooneandthesamevectorbasisresultingfromtheidentificationoftangentvectorbasesatdifferentpoints.Wethushaveinthoseconstantfieldsdvi=vi.(10.7)∂RRItthenfollowsthat,ifdv=0,0=vi(B)−vi(A).(10.8)Thisisthecaseforanypairofpoints.Thusviisaconstantandsoisv.Oncethisresulthasbeenobtained,wemaytakearbitrarybasisfieldstoexpresstheconstancyofv,i.e.vie=vie,(10.9)iAiBthoughthisequalitydoesnotholdforthecomponentsthemselves.Sincetheequationdv=0isequivalenttovi=0,theconservationlaw,(10.6),may;jbesaidtobeaconsequenceofvi=0,butonlyifthereisteleparallelism.If;jtherearenoconstantframefields,theequationdv=0isdeceitful,sinceitismeaningless.

1818.10.TENSOR-VALUEDNESSANDTHECONSERVATIONLAW163Similarly,letTbeatensorfieldsuchthatdT=0.Ifandonlyifthegeometryhasteleparallelism(Kleingeometriesinparticularhaveit),wemayconcludethatT(A)=T(B)(10.10)foranypairofpoints(A,B).Inotherwords,Tisaconstanttensorfield.But,aswasthecasewithvectorfields,(10.10)doesnotevenmeananythingwithoutteleparallelism.Onecantrytosavetheconservationlawwhenthereisnotteleparallelismbyconfiningoneselftoverysmallregions.Inthatcase,wecanfollowCartanindiscussingthespecificcaseofavector-valueddifferential3−formτinspacetime[8](hewasdiscussingenergy-momentumtensors,whicharedisguisedformsofsuchdifferentialforms).Afterdefiningitsexterior(-covariant)derivative,dτ,hestatedthatτisconservedifdτ=0,inwhichcaseτ=0.(10.11)∂RButnon-scalar-valuedintegrationisnotwithoutproblemsevenforsmalldo-mains,aswenowexplain.CartanidentifiedthefirstBianchiidentitywiththestatementofconservationofthetorsion.ButthefirstBianchiidentitydoesnotstatethattheexterior(-covariant)derivativeofthetorsioniszerounlesstheaffinecurvatureiszero.So,theconservationofthevector-valueddifferentialformtorsionprovidesaverytellingexampleofhowintrinsicallylimitedisanystatementofconservationofvectorandtensor-valuedformswhenthereisnotteleparallelism.Therelevancethatthecurvaturehasindeprivingtheconservationofnon-scalar-valueddifferentialformsoftruemeaningmanifestsitselfinparticularinconnectionwiththetorsion,i.e.theformalexterior(-covariant)derivativeofdP.Wewouldthinkthat,whenthetorsioniszero,theintegralofdPontwodifferentpathsbetweenthesametwopointsmustbeequal:BBdP=dP.(10.12)A;(path1)A;(path2)Butwesawwithsimpleexamplesinthepreviouschaptersthatthisisnotcorrect.Strictlyspeaking,thedevelopmentofaclosedcurvedoesnotcloseingeneral,independentlyofhowsmallthecurveis.ReturningtothepointwepreviouslymadethattheconservationlawhastodowithexteriorderivativesinthesenseofCartanandK¨ahlerandnotwithcovariantderivatives(unlesstheycoincide,asisthecasewithvectorfields),theconservationoftheEinsteintensorwouldseemtocontradictit.Weshallseeinalaterchapterthatthisisaveryspecialcase.IthappensthattheannulmentofthecomponentsofthecovariantderivativeoftheEinsteintensorisequivalenttotheannulmentofthecomponentsoftheexterior(-covariant)derivativeoftheEinsteinvector-valueddifferential3−form,whichiswhatthatfamousmathematicalobjectreallyis.Butcovariantderivativesarenotrelevant

182164CHAPTER8.AFFINECONNECTIONSingeneral.Forinstance,thevanishingoftheexterior(-covariant)derivativeofthecurvature,jid(Ωie∧ej)=0,(10.13)doesnotcontainthesameinformationasthevanishingofitscovariantderivativesjR=0(10.14)ikl;s(Seesection12ofthischapter).Thosetrivialobservationsarenecessaryinordertoidentifymisstatementsabouttheconservationlawingeneralrelativity.Ifoneperformssuchillegalintegrationsandonegetstherightresults(meaningthattheyappeartoexplainphysicalobservations),itmayhappenthattheconnectionofspacetimeisateleparallelone,ratherthanthespecificonethatispresentlyascribedtoitandwhichisknownasLevi-Civita’s.Teleparallelismmaybeassumedinadvertentlywhenonemakesaspecificbasisfieldplayaspecialroleinacomputation.Inchapters9and10weshalllearnabouttherelationbetweenteleparallelconnectionsandLevi-Civita’s.8.11Thezero-torsioncaseWhenthetorsioniszero,thefirstequationofstructureandthefirstBianchiidentitiesrespectivelybecomedωi=ωj∧ωi(11.1)jji0=ω∧Ωj.(11.2)Equation(11.2)impliesRi+Ri+Ri=0,(11.3)jklkljljkwhichisvalidinarbitrarybases.Theuseofcoordinatebasesmakesthelefthandsideof(11.1)disappearandthefirstequationofstructurebecomeΓi=Γi(11.4)kjjkintermsofthosebases,indicatedbyprimes,butnotingeneral.Thepropertyofzerotorsionofaspacetakesaparticularlyinterestingformintermsofintersectingfamiliesofcurves.TherepresentationPP+PM+MM+MPofPPMMP(Fig.7)canberewrittenas(PM−PM)−(MM−PP).(11.5)LetΔλandΔubesmallquantities.Then∂(nΔλ)∂nPM−PM=Δu=ΔλΔu(11.6a)∂u∂u

1838.12.HORRIBLECOVARIANTDERIVATIVES165∂(tΔu)∂tMM−PP=Δλ=ΔλΔu.(11.6b)∂λ∂λTheseareconnectionindependentequations.BothMM−PPandPM−PMgotozeroasΔuandΔλdo.However,PM−PMMM−PPandΔuΔλΔuΔλdonot.ThestatementthatPP+PM+MM+MPiszerooninfinitesimalcurvesmeansthat∂n−∂tgoestozerointhelimitand,therefore,∂u∂k∂n∂t=.(11.7)∂u∂λTakingthisto(7.10)makesitbecome∂2n=tiRjtknle.(11.8)∂u2ikljNoticethat,asdefinedinsection6,nisthevectorassignedtoan“instanta-neous”changeoftheparameterλbyoneunit.Theterminstantaneousisusedintheabsenceofabetterword,meaningfortheparameterλwhatitliterallymeanswhentheparameteristime.Wehavederivedequation(11.8)inapurelyaffinecontext.ItalreadylooksliketheformulaforgeodesicdeviationinRiemanniangeometry.Butwehavenotyetconsideredthatgeometry;metricshavenotbeenusedinobtaining(11.8).8.12HorriblecovariantderivativesWeshallnowexpresstheBianchiidentities(i.e.thedifferentiatedequationsofstructure)intermsofcomponentsofcovariantderivativesasanexampleofhowcumbersomesuchexpressionsbecome.Torsionandcurvatureare,ofcourse,nottensors.TheircovariantderivativeswouldbethecomponentsoftheCartan-K¨ahlerexteriorderivativeiftheyweretensorsofrespectiveranks3and4.Buttheyarerather(1,0)-valuedand(1,1)-valueddifferential2−forms.ThefirstBianchiidentityinvolvesbothcurvatureandtorsion.WethusstartwiththesecondBianchiidentity,dΩj+Ωi∧ωj−ωk∧Ωj=0,(12.1)ikkikjsinceitinvolvesonlycurvature.IfwegiveΩiintermsofacoordinatebasisof2−forms,i.e.as1Rjdxk∧dxl,thereisindΩjonlythetermresultingfromthe2iklidifferentiationofRj.If,ontheotherhand,werepresentitas1Rjωk∧ωl,ikl2ikljthreetermsresultfromdΩi,foratotaloffive,onthelefthandsideof(12.1).Onlythelasttwotermsinthisfirstdevelopment,whicharethesamelasttermsasin(12.1),involvetheconnection.

184166CHAPTER8.AFFINECONNECTIONSjOntheotherhand,thecovariantderivativeRhasfivetermsofwhichikl;mjfourdependonconnectionsinceRisthecoefficientofthe(1,3)-valueddif-ikl;mjikljferential1−formd(Riklφφφej).InordertomakeRikl;mappearin(12.1),weinvokethefirstequationofstructuretosubstitutedωkanddωlinthedif-ferentiationof1Rjωk∧ωl.Wethusobtainseventermsonthelefthandside2iklof(12.1).FiveofthetermsyieldRjωm∧ωk∧ωl.Thetwoextratermsareikl;m1jhl1jkhjhl1jhmklRihlΩ∧ω−Rikhω∧Ω=RihlΩ∧ω=RihlRmkω∧ω∧ω.(12.2)222Equation(12.1)thusbecomes(Rj+RjRh)ωm∧ωk∧ωl=0.(12.3)ikl;mihlmkWecompletethetranslationtothetensorcalculusbyexpressing(12.3)intermsofcomponents:(Rj+RjRh)+(Rj+RjRh)+(Rj+RjRh)=0.(12.4)ikl;mihlmkilm;kihmklimk;lihklmThereare36indicesin(12.4)versusonlytenin(12.1).Theremaybeadis-crepancybyafactoroftwowithmuchoftheliteratureinthesecondofthetwotermsineachparenthesis.Thedifferencevanishesinonedefinesthecompo-nentsofthetorsionbyRhωm∧ωkratherthanbyRh(ωm∧ωk).ThenewmkmkRhwouldbehalfwhattheoldoneis,whichwouldcause(12.4)tobecomemk(Rj+2RjRh)+(Rj+2RjRh)+(Rj+2RjRh)=0.(12.5)ikl;mihlmkilm;kihmklimk;lihklmAssumethatthetorsionwerezero.Eventhentheconservationlawofthecurvaturewouldnotbegivenbytheannulmentofitscovariantderivative,butbyalessrestrictivecondition,as(12.5)immediatelyshows.Allthisspeaksofhowinimicalcovariantderivativesaretotheissueofconservation.ConsidernowthefirstBianchiidentitydΩi+Ωj∧ωi−ωj∧Ωi=0.(12.6)jjWewritethetorsionasi1ijkΩ=Rjkω∧ω.(12.7)2Whenwedifferentiate(12.7),weshallhavefivetermsontherighthandsideifweusethefirstequationofstructuretosubstitutefordωjanddωk.ThreeofthosetermstogetherwithΩj∧ωiformj1iljkRjk;lω∧ω∧ω.(12.8)2TheothertwotermsarisingfromsubstitutingdωjanddωkonthedifferentiatedΩiresultin1ihk1ihjRhkΩ∧ω−RjhΩ∧ω(12.9)22

1858.13.AFFINECONNECTIONS:RIGOROUSAPPROACH167andcanbewrittenfurtherasihk1ihljk...=RhkΩ∧ω=RhkRljω∧ω∧ω.(12.10)2Thelasttermin(12.6)is1iljk−Rljkω∧ω∧ω.(12.11)2Whenwesubstitute(12.8),(12.10)and(12.11)in(12.6),weget(Ri+RiRh−Ri)ωl∧ωj∧ωk=0,(12.12)jk;lhkljljkand,therefore,(Ri+RiRh−Ri)+(Ri+RiRh−Ri)+(Ri+RiRh−Ri)=0.jk;lhkljljkkl;jhljkjkllj;khjklklj(12.13)DefiningagainthecomponentsofthetorsionasRhωm∧ωkratherthanasmkRh(ωm∧ωk),wewouldget,insteadof(12.13),mkiihiiihi(2Rjk;l+4RhkRlj−Rljk)+(2Rkl;j+4RhlRjk−Rjkl)++(2Ri+4RiRh−Ri)=0.(12.14)lj;khjklkljNoticethatthereare42indiceshere,tobecomparedwiththesevenindicesin(12.6).Noticealsothat,evenifthecurvaturewerezero,theconservationoftorsionwouldnotbegivenbytheannulmentofitscovariantderivative.8.13Affineconnections:rigorousAPPROACHWestartbyrecallinghowLiealgebrasofaffinegroupsemergeinthecontextofthemovingframe.Recalltheaffinegroup:P=Q+Aia,e=Aja,whereiiij(Q,a1...an)isafixedframe.FromitweobtainedijdP=ωei,dei=ωiej,(13.1)withωi=dAmBi,ωj=dAmBj.(13.2)miimInotherwords,dg·g−1isthe1−formofcomponents(ωi,ωj).Thedifferentialiformsdωi−ωj∧ωianddωj−ωk∧ωjalsoaregoingtoplayaprominentrolejiikinthetheoryofaffineconnections,butwenolongerhave0=dωi−ωj∧ωi,0=dωj−ωk∧ωj.(13.3)jiikWeexceptionallyusetheterm“affine-linear”ratherthansimplyaffineasameanstoexcludetheaffineconnectionsofthemoregeneralFinslertype,whichcanbedefinedinasimilarbutslightlymorecomplicatedway[73].

186168CHAPTER8.AFFINECONNECTIONSDefinition1Anaffine-linearconnectionisaone-form(ωi,ωj)onan(n2+n)-idimensionalmanifoldB(M),takingvaluesintheLiealgebraoftheaffinegroupfordimensionnandsatisfyingtheconditions(1)Then2+nreal-valuedone-formsarelinearlyindependent.(2)Thedifferentialformsωiarethesolderingforms,meaningthattheyarethecoefficientsofthetranslationdifferentialformdPwhenpushedforwardfromasectiontotheframebundlebytheactionofthegenerallineargroupGL(n).j(3)ThepullbacksofωitothefiberofthefibrationΠ:B(M)→MaretheleftinvariantformsofGL(n).(4)TheformsΩi=dωi−ωj∧ωi,calledtorsion,andΩj=dωj−ωk∧ωj,jiiikcalledaffinecurvature,arequadraticexteriorpolynomialsinthenformsωiΩi=Riωj∧ωk(Ri+Ri=0)(13.4)jkjkkjΩi=Riωk∧ωl(Ri+Ri=0).(13.5)jjkljkljlkTheorem8.1Theintegrationofthesystemofdifferentialequations(13.1)alongacurveinB(M)givesanaffinetransformationbetweentangentspacestoM.Proof.Onparametrizationsf(u),(a≤u≤b)ofcurvesinB(M)(weas-sumethatfissufficientlysmooth),the(ωi,ωj)arenowthepullbacksof(ωi,ωj)iiinto[a,b].Becausetheequationsarelinear,thesolutionislinearwithrespecttotheinitialconditionsand,therefore,takestheform(P=A+fi(u)a,e=gja),iiijwithA=P(a)andai=ei(a)asinitialconditions.Butei(a)istheinitialbasis,i.e.thebasisf(a).P(a)istheoriginofthetangentvectorspaceatΠ(f(a)).SimilarinterpretationsapplytoP(u)andei(u)atΠ(f(u)).Hencewehaveob-tainedanaffinetransformationbetweenthevectorspacestangentatthepointsΠ(f(a))andΠ(f(u)).P(u)describesacurveinthetangentvectorspaceatΠ(f(a))calledthedevelopmentofthecurvefintothatvectorspace.Theorem8.2Theintegrationoftheconnectionisindependentofthesectionusedtoperformit.Proof.Letg(u),(a≤u≤b)beacurveinM.TakeanothercurveG(u),(c

1878.13.AFFINECONNECTIONS:RIGOROUSAPPROACHAFFINECONNECTIONSAFFINESPACEAFFINECONNECTIONSμP=Aaμνeμ=Aμaνμλω,ωνdP=ωμe∗μμd¯P=ωeμdeν∗νμ=ωμeνd¯eμ=ωμeνdv=(dvμ+vνωμ)eμd˙v=(dvμ+vνωμ)eννμdωμ−ων∧ωμ=0d˙d¯P=Ωμ∗eμΩμ=dωμ−ων∧ωμννdων−ωλ∧ων=0d˙d¯eν∗Ων=dων−ωλ∧ωνμμλμ=Ωμeνμμμλddv=0d˙d˙v=vμΩνeνμ0=0dΩμ=−Ων∧ωμ+ων∧Ωμνν0=0dΩν=−Ωλ∧ων+ωλ∧Ωνμμλμλ∗d¯isnotanoperatorbutpartofsymbols¯dPand¯de;d˙=formaldifferentiation;d˙d˙=0;Ωμ≡Rμ(ων∧ωλ),Ων=Rν(ωλ∧ωπ)νλμμλπ169

188May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

189Chapter9EUCLIDEANCONNECTIONS9.1MetricsandtheEuclideanenvironmentThetermEuclideanconnectionisduetoCartanandpertainstothecomparisonofEuclideanneighborhoodsatdifferentpointsofadifferentiablemanifold.ItisabettertermthanthemoreusualoneofRiemannplustorsiongeometry,whichreflectsthehistoricalaccidentthattherewasRiemanniangeometrybeforethebirthoftheconceptoftorsion(otherthantorsionofcurves,whichissomethingdifferent).Torsionisapropertyofconnections.Euclideanspacehasaconnection,theonethatmakesi,jandkbethesamethreevectorseverywhereinEuclideanspace.Itthushasaconceptoftorsionthathappenstobezero.Incontrast,theoriginalRiemanniangeometrydidnothaveaconceptofcomparisonofEuclideanneighborhoods,anddidnot,therefore,haveaconceptoftorsion,notevenimplicitly.Whatwoulda“non-Euclideanenvironment”meanwhenspeakingofmet-rics?ItwouldmeanaconceptofmetricasinRiemanniangeometryinthefirstepochofitsexistence.Cartanreferredtothosemanifoldswiththeterm“falsespacesofRiemann”[13].Asheexplained,theybecametruespacesin1917,withtheadventinthatyearoftheLevi-Civitaconnection(LCC)[52].Itmustbesaid,however,thatconnectionswerepresentinacrudeimplicitformintheworkofRicciandLevi-Civitain1901[60].WeshalldevotethefirsttwosectionsofthenextchaptertothefalsespacesofRiemann,i.e.withoutacon-nectionand,therefore,withoutaEuclidean(norpseudo-Euclidean)structure.Inthemeantime,letusspendacoupleofparagraphstoprovidealittlebitofbackgroundonmanifoldswithexclusivelymetricstructure.Assumewehadadifferentiablemanifoldendowedwithametricds2=g(x)dxμdxv(1.1)μv171

190172CHAPTER9.EUCLIDEANCONNECTIONSbutwithoutaconnection.Onecouldobtainsetsofωμ’sthatorthonormalizeit,i.e.suchthatds2=ηωμωμ,(1.2)μμwhereημ=±1.TakeoneofthosesetsandpushittotheEuclideanframebundlebytheactionofG0(itisthegroupofrotationsinndimensions,ifthesignatureisEuclidean;anditisthegroupconstitutedbyboosts,rotationsintheEuclideansubspaceandtheproductsofallthose,ifthesignatureisLorentzian).Considerthesystemofdifferentialequationsμμdy−ω(x,u,dx)=0,(1.3)whereurepresentstheparametersinG0.Thissystemmayadmitornotsolu-tionsforyμ.Ifitdoes,themetric(1.1)canberewrittenasds2=η(dyμ)2,(1.4)μμημbeing±1.Anecessaryandsufficientconditionforexistenceofasolutionto(1.3)isthatthesetofquantitiesnowadaysknownascomponentsoftheRiemanniancurvaturebezero.Theyarethusananswertoaspecificproblemofintegrabilitywithoutreferencetoacomparisonofvectors,whichdonotentertheargument.Buttherestillisaconceptofbundle(implyingaspecificdependenceoncoordinatesabsentfromthemetric)wherethegroupinthefibers(rotations,orrotationstogetherwithhyperbolicrotations)isthesameasintheEuclideanbundleofframesforthesamedimension.Whenonetriestoaddaffinestructuretothatmanifold,theissueofcom-patibilityofthetwostructuresarises.Thisisnot,however,Cartan’sapproachtoEuclideanstructure,generalizedornot.Forhim,Euclideangeometry,andpost-1917Riemanniangeometry,isfirstandforemostaboutconnectionsthatarerestrictionsofaffineconnections.Thismeansthatvectorbasesarereplacedwithorthonormalorpseudo-orthonormalones.EndofconsiderationsonRie-mannianpseudo-spacesinthischapter.AsinthegeometryofRiemannianpseudo-spaces,thegroupG0inthefibersisthegroupofisometries(rotationsinEuclideanspaces,andLorentztransfor-mationsinthespacetimeofspecialrelativity),bothinEuclideanspacesandintheirCartangeneralizations.OnceaEuclideanframe(i.e.wherethevectorbasisisorthonormalorpseudoorthonormal)hasbeenchosen,thepointsineachfiberareinaonetoonecorrespondencewiththemembersofG0.Theroleofthepair(G,G0)indefininggeometriesadvisesagainsttheuseoftheterm“metriccompatibleaffineconnection”,sothatwedonotoverlookthatrole.Laterinthischapter,weshallstarttodealwithpseudo-Euclideandiffer-entiablemanifolds,i.e.wherethesignatureofthemetricispseudo-Euclidean,andspecificallyLorentzian.Correspondingly,weshalluseGreekindicestogofrom0ton−1.Latinindiceswillgofrom1ton−1.Thenotationμ=(0,m)willthenbeobvious.

1919.2.EUCLIDEANSTRUCTUREANDBIANCHIIDENTITIES173LetusreviewhowtheconceptsofChapter6havetobeunderstoodwhenthemetricneitherisofthetype(1.4),norcanbereducedtoit.Regardlessofaffineconnection,onedefinesds2asdP·dP.Butoneshouldratherwriteds2=dP(⊗,·)dP=g(x)dxμ⊗dxν,(1.5)μνwherethefirstsymbolin(⊗,·)ismeanttorefertothetensorproductinthealgebraofdifferentialforms,symbolwhichisusuallyoverlooked.Moregenerally:ds2=ωμe(⊗,·)ωνe=gωμ⊗ων.(1.6)μνμνwheregμν=eμ·eν.(1.7)Fororthonormalandpseudo-orthonormalbases,gμνisdiagonalandgμμequalsημ,meaning±1.Themetriccanthenbewrittenas2μμds=ημω⊗ω,(1.8)μwhichisamoreinformativewayofwriting(1.2).ds2isnotanexteriordifferential2−form,or,forthatmatter,anexteriordifferentialr−formforsomer.Oneintegratesωμωvgμv(x)⊗dλ(1.9)dλdλoncurves.Thenotationωμ/dλisjustifiedbytheproportionalityofωρtodλinonedimension(i.e.oncurves).Ofcourse,thetensorproductinexpression(1.8)isunnecessarysincetheωρ/dλarescalarvalued.TheproperwayofwritingthemetricintheEuclidean(orpseudo-Euclidean)framebundleofadifferentiablemanifoldendowedwithametricisgivenby(1.2),equivalently(1.8);ontheotherhand,(1.1),equivalently(1.5),belongstotheaffineextension(seenextsection)oftheEuclideanbundle.Ourstartingpointinthissectionshouldhavebeen(1.8),wereitnotforourrespectingtraditiononthismatter.9.2EuclideanstructureandBianchiIDENTITIESAEuclideanframebundleofadifferentiablemanifoldistherestrictionofitsaffineframebundlefromvectorbasestoorthonormalorpseudo-orthonormalbases;oneisreplacingapair(G,G0)withanother.Atthispoint,knowledgeablereaders(andalsootherswhoarenotsomuchsobutareinahurry)mightwhichtojumptotheequationsofstructure,(2.9)to(2.11),toseethatwearejustaddingEq.(2.11)totheequationsofstructureofthepreviouschapter.

192174CHAPTER9.EUCLIDEANCONNECTIONSWhereastheaffineframebundleisofdimensionn+n2,theEuclideanframebundleisofdimensionn+[n(n−1)/2].Hence,ifandwhenonesaysthataEuclideanconnectionisarestrictionofanaffineconnectionfromanaffineframebundletoaEuclideanframebundle,onemeansthataEuclideanconnectionistoaEuclideanframebundlewhatanaffineconnectionistoanaffineframebundle.Oneusestheterm“extensionofanEuclideanconnectiontoanaffinecon-nection”when,bytheactionofthelineargroupforthesamedimension,oneextendstheformulasforaEuclideanconnectionfromaEuclideanframebundletoanaffineframebundle,i.e.wheretheframesarenotorthonormal.Differentiationof(1.7)yieldskkdgμν−ωμgkν−ωνgkμ=dgμν−ωμν−ωνμ=0(2.1)fortheaffineextensionofaEuclideanconnection.Recallingalreadydefinednotationfortheslashsubscript,wehavedg=gωλ.(2.2)μνμν/λWedefineΓ’sandΓ’sascomponents,respectivelyintermsofωλ’sandμνλμνλofdxλ’s,ofpull-backsofωtosectionsofthebundle:μνg−Γ−Γ=0,(2.3)μν,λμνλνμλandgμν/λ−Γμνλ−Γνμλ=0.(2.4)Hence,neithertheωnortheΓ’sandΓareskew-symmetricunlesstheμνμνλμνλbasesareorthonormalorpseudo-orthonormal,inwhichcasedgμνiszero.Theequationsωμν+ωνμ=0.(2.5)andΓμνλ+Γνμλ=0,Γμνλ+Γνμλ=0.(2.6)thenfollow.IntheproperlyEuclideancase(positivedefinitemetric),equation(2.5)uclideanimpliesων=−ωμ,(2.7)μνand,intheLorentziancase,ωi=ω0,ωj=−ωi.(2.8)0iijTheΓ’sarecalledChristoffelsymbols.Thereisnoneedtoretainthisname,μνλexceptforbeingabletorecognizewhatitrefersto,iffoundintheliterature.Equations(2.1)-(2.4)belongtotheextensionofanEuclideanconnectiontoanaffineconnection.Equations(2.5)-(2.8)belongtotheEuclidean(orpseudo-Euclidean)connectionsthemselves.Werepeatthattheextensionisnotanaffineconnectionproper.

1939.2.EUCLIDEANSTRUCTUREANDBIANCHIIDENTITIES175BeforeproceedingwiththedefinitionofEuclideanconnection,weadviseinexperiencedreaderstorevisitthelastsectionofchapter6,wheretheLiealgebraoftheEuclideangroupisintroduced.Definition2AdifferentiablemanifoldMendowedwithametricissaidtobeendowedwithaEuclideanconnectionifitsEuclideanframebundleB(M),whichisofdimensionn+1n(n−1),isendowedwithadifferentialone-form(ωμ,ων)2μtakingvaluesinthe(real)LiealgebraoftheEuclideangroupandsatisfyingtheconditions:(1)Then+1n(n−1)real-valuedone-formsarelinearlyindependent.2(2)Theformsωμarethesolderingforms,meaningthattheyarethecoeffi-cientsofthetranslationdifferentialformdPwhenpushedforwardfromanysectionofB(M)toB(M)itself.(3)ThepullbacksofωνtothefiberofΠ:B(M)→MaretheleftinvariantμformsofO(n),oroftheLorentzgroupifthesignatureis(1,−1,...,−1).(4)TheformsΩμ=dωμ−ων∧ωμ,calledthetorsion,andΩν=dων−ωk∧ων,νμμμkcalledtheEuclideancurvature,arequadraticexteriorpolynomialsinthenformsωμΩμ=Rμ(ων∧ωλ)(Rμ+Rμ=0)(2.9)νλνλλνandΩμ=Rμ(ωλ∧ωπ)(Rμ+Rμ=0).(2.10)ννλπνλπνπλTotheseequationsofstructure,weaddωμν+ωνμ=0,(2.11)whichisthedifferentialformof(1.7).Thisisconsistentwithcondition(3).ItiseasytoshowfromΩν=dων−ωk∧ωνand(2.11)thatμμμkΩμν=−Ωνμ,(2.12)whichwasnotthecaseforaffineconnectionsingeneral.Althoughwehaveprovedthisequationinorthonormalbasisfields,thisisclearlyvalidalsofortheaffineextension.WedonotspeakofEuclideanandaffinetorsionssince,unlikethecaseofthecurvature,thetorsionretainsthesamenumberofindependentcomponentsinbothcases.IntheEuclideanenvironment,theconceptofEuclidean-Kleingeometryisfirst,followedbytheconceptofEuclideanconnection,thenfollowedbytheconceptofmetricasaderivedinvariantds2=η(ωμ)2,meaningηωμ⊗ωμ,(2.13)μμμμ

194176CHAPTER9.EUCLIDEANCONNECTIONSThemetricallowsustoreplacealinearfunctionofvectorswithavector,theactionofthelinearfunctionbeingreplacedwiththedotproductwiththatvector.WemaythuswritetheEuclideancurvatureas≡Ωνeμ∧e,(2.14)μνinsteadof(8.3)ofchapter8,andtheBianchiidentitiesasdΩ=ωj∧Ωie,d≡0.(2.15)jiWhenthetorsioniszero,theEuclideanconnectioniscalledtheLevi-Civitaconnection(LCC),denotedαν.Weshallseeinthenextsectionthatitdependsμonlyonthemetricanditsfirstderivatives.Thecurvaturedαν−αλ∧ανreceivesμμλthenameofRiemann’scurvature.HenceRiemanniancurvaturemeanstwothings.Ontheonehand,itisthesetofquantitieswhosebeingzeroconstitutestheintegrabilityconditionof(1.3);itconcernsdifferentiablemanifoldsregardlessofwhethertheyareendowedwithaconnectionornot.But,ifthemanifoldisendowedwiththeLevi-Civitaconnection,itisatthesametimetheEuclideancurvature,playingtherolethataffinecurvaturesplayinmanifoldsendowedwithaffineconnection.Wewrite≡Ωνeμ∧e=Rμ(ωλ∧ωπ)eμ∧e,(2.16)μννλπνΩν≡dαν−αλ∧αν,(2.17)μμμλtomakeexplicitthemeaningofdifferentsymbols.Whenthetorsionisnotzero,theEuclideanconnectionofthedifferentiablemanifoldnolongeristheLCC.WestillmaydefinesymbolsανandΩνthatwillμμtakethesameformasifthetorsionwerezero,exceptthatweshallnolongeruseforthemthetermsconnectionandEuclideancurvature,becausetheynolongerareso.Theyareconceptsderivedfromthemetric,equivalentlyfromthefundamentalinvariantsωμ’sthatparticipateindefiningtheEuclideanstructure.Invariantsderivedfromthefundamentalonesneednotinvolveallofthelastonesatthesametime.Thetwocurvatures,EuclideanandRiemannian,arerelatedthroughthetorsionanditsderivatives,asweshallseeinsection8.OfspecialinterestisthecasewhentheEuclideancurvaturesatisfiesωμ∧Ων=0,(2.18)μliketheRiemanniancurvaturedoes.Butalsodoallthoseconnections,affineorEuclidean,whoseexterior(-covariant)derivativeiszerodΩ=0.(2.19)Equation(2.19)impliesRν+Rν+Rν=0.(2.20)μλπλπμπμλ

1959.3.THETWOPIECESOFAEUCLIDEANCONNECTION177ApropertyspecifictoEuclideanconnectionsisRμνλπ=Rλπμν.(2.21)forEuclideancurvatures.Inordertoprove(2.21),welowertheindexνinEq.(2.20).WerewritetheresultingequationasRμ(ν)λπ=0,(2.22)whichdefinesRμ(ν)λπ.Weaddto(2.22)threemorecopiesofthisequationwithexchangedindicestogetRμ(ν)λπ+Rν(μ)λπ+Rλ(μ)πν+Rπ(μ)νλ=0.(2.23)Wedevelopeachofthefourtermsonthelefthandsideaspertheirdefinitionandusetheskew-symmetryunderexchangeofthelasttwoindices.Termscanceloutandwefinallyget(2.21).Becauseofcurrentusageintheliterature,weshallsometimesspeakofaffineconnectionswhenreferringtoextensionsofEuclideanconnections,andeventotheEuclideanconnectionsthemselves.Indoingso,onestillmakestheimplicitpointthatthemetricstructureissubordinatedtotheaffinestructure,thencalledEuclidean.9.3ThetwopiecesofaEuclideanconnectionSometimes,wemaynotknowthesetoffundamentalinvariants(ωμ,ων)ofaλdifferentiablemanifold,butwemayknowΩμandωμ.ThesystemΩμ=dωμ−ων∧ωμ,(3.1)ν0=ωμν+ωνμ(3.2)thendeterminesωμ.νWhenthetorsionisnotzero,wemaystilldefineLevi-Civitasymbols,αμ,νasthosewhichsatisfythesystemμνμ0=dω−ω∧αν,(3.3)0=αμν+ανμ.(3.4)WedonotrefertoαμastheLCCinthiscase.Definingνβν≡ων−αν,(3.5)μμμallowsustoviewαandβastwocomplementarypiecesofEuclideanconnections.νOnereadilyshowsthatβμsatisfiesthesystemΩμ=−ων∧βμ.(3.6)ν

196178CHAPTER9.EUCLIDEANCONNECTIONS0=βμν+βνμ.(3.7)βνand,betteryet,β(=βνeμ∧e)arecalledcontorsion.WeintroducenewμμνsymbolsLνbyμλβ≡βνeμ∧e≡Lνeμ∧eωλ.(3.8)μνμλνWenowreviewtheissueoftensoriality,equivalently,horizontality.dPistheunitvector-valueddifferential1−formdP=δνωμe.(3.9)μνItisaninvariant;theωμaretensorial.Ω,definedasd(dP),isaninvariantμbecausesoaredanddP.ThecomponentsRofΩ,definedbyνλμνλ1μνλΩ=Rνλ(ω∧ω)eμ=Rνλω∧ωeμ,(3.10)2aretensorial.Wealsosawthenon-tensorialtransformationofωνandαν.Theirμμdifference,βμ(=ων−αν),however,istensorialsincethenontensorialtermsνμμcancelout.Clearly,wehavesμsνβλ=AλAνβμ,(3.11)and,equivalently,βνeμ∧e=βνeμ∧e=βνeμ∧e=....(3.12)μνμνμνBeawareofthefactthatαμνe∧earenotsimilartoβμνe∧eonsectionsμνμνoftheEuclideanframebundlesincetheανandωνdonottransformlinearlyμμunderthegroupofisometries.Weproceedwithaclarificationofthemeaningofeμ.Whereaswehaveμμde1=ω1(x,dx)eμonasection,wehavede1=ω1(x,u,dx,du)eμinthebundle.Thisemphasizesthat,ateachpointxofthebasespace,e1doesnotrepresentavector,butthefirstelementofallpossible(orthonormal)basesatx.9.4ComputationincoordinatebasesoftheaffineextensionoftheLevi-CivitaconnectionTheaffineextensionoftheequation(3.4)satisfiedbytheLCCisdgμν=αμν+ανμ,(4.1)whichimplies,usingprimesfortheimplicitbasesofdifferential1−forms,ππgμν,λ=Γμλgπν+Γνλgπμ.(4.2a)Weshalluseunderlinedquantitieswheneverwerefertothecomponentsofαorofitscurvature.Weinterchangethesubscriptsμandλtoobtainππgλν,μ=Γλμgπν+Γνμgπλ.(4.2b)

1979.5.COMPUTATIONOFTHECONTORSION179andthesubscriptsνandλtogetππgμλ,ν=Γμνgπλ+Γλνgπμ.(4.2c)Weaddequations(4.2a)and(4.2b),subtract(4.2c)andusethesymmetryππΓμν=Γνμ,(4.3)whichisequation(11.4)ofthepreviouschapterappliedtotheLCC.Wethusgetπgμν,λ+gλν,μ−gμλ,ν=2Γμλgπν.(4.4)Withtheappropriatecontraction,onearrivesatthesoughtresult,namelyπ1νπΓμλ=g(gμν,λ+gλν,μ−gμλ,ν),(4.5)2whichshowsbyexplicitcalculationthatasolutionexistsandisunique.Exercise.FindtheΓπ’sforthelineelementds2=dP·dP=dρ2+ρ2dθ2.μλVerifythattheEuclideancurvatureiszero.Exercise.Sameproblemforthelineelementsds2=dP·dP=dρ2−ρ2dθ2(4.6)andds2=dr2+r2dθ2+r2sin2θdφ2.(4.7)Exercise.FindtheLevi-Civitaconnectionfords2=e2ν(r)dt2−e2λ(r)[dr2+r2(dθ2+sinθdφ2)],(4.8)whereν(r)andλ(r)areundeterminedfunctionsthatadmitsecondderivatives.Thesolutiontothisexerciseistediousthroughtheuseof(4.5).WeshalllaterseeafasterwaytosolvefortheLevi-Civitaconnection(byinspection,whichisadvantageousinmostcases).9.5ComputationofthecontorsionWeproceedtocomputethecontorsionofEuclideanconnections.Werewrite(3.6)asΩ=−ων∧β,(5.1)μνμandrecallthedefinitionofthecomponentsofβνμthroughβ=Lωλ.(5.2)νμνμλ

198180CHAPTER9.EUCLIDEANCONNECTIONSInthisway,wehaveνλνλΩμ=−Lνμλω∧ω=(Lλμν−Lνμλ)(ω∧ω),(5.3)where,asusual,theparenthesisin(ων∧ωλ)isusedtosignifysummationoverabasisof2−forms,ratherthanoverallpossiblevaluesoftheindicesνandλ.Recallthatwedefinedthecomponentsofthetorsionby1νλνλΩμ=Rμνλω∧ω=Rμνλ(ω∧ω).(5.4)2Noticethatthecoefficientsof(ων∧ωλ)in(5.4)arenotdifferencesofR’s,unlikethecoefficientsontherighthandsideof(5.3).Weequatecorrespondingcoefficientsin(5.3)and(5.4)andgetRμνλ=Lλμν−Lνμλ=Lμνλ+Lλμν,(5.5)wherewehaveusedtheskew-symmetryofβ,andthusofL,withrespecttothefirstpairofindicesinLνμλ.TwomorecopiesofthisequationwiththeindiceschangedareRνλμ=Lνλμ+Lμνλ(5.6)Rλνμ=Lλνμ+Lμλν.(5.7)AddingthelastthreeequationsandtakingintoaccountthatLπσρ=−Lσπρ,wehave1Lμνλ=(Rμνλ+Rνλμ+Rλνμ).(5.8)2Amnemonicruleforthisequationis:formthecyclicsumofR’sstartingwiththesameindicesμνλasonthelefthandside,andreplaceRλμνwithitsnegative,νRλνμ.Wegetβμasβν=ηβ=ηLωλ.(5.9)μνμννμνλwhereηνis±1,dependingonthesignatureofthemetric.Intheprevioussection,wecomputedtheLCCintermsofcoordinatebases.Ifwewishtocomputeitintermsoforthonormalbases,sufficetonoticethatαsatisfiesasystemofequationssimilartoβ’s.Since−dωμ=−ων∧αμ,theνcomputationthatwecarriedoutinthissectioncanberepeatedwith−dωμplayingthesameroleasΩμ.Donotoverlooktoskew-symmetrize−dωμ,i.e.toexpressitintermsofbasis(ωμ∧ων).9.6ComputationoftheLevi-CivitaconnectionbyinspectionWenowsolvefortheLCCbyinspection,usingorthonormalbases.Itisaconvenientmethodwhenthemetricisnotverycomplicated.Mostinterestingcasesarelikethat.Readersarewarnedthat,inordertoavoidalengthytitleofthesection,wehavereferredinittotheLCC.Whatweareabouttocompute

1999.6.LEVI-CIVITACONNECTIONBYINSPECTION181isαν,whichmayormaynotbetheEuclideanconnectionofthemanifold,butμwhichalsoispresentwhenthemetricis.Onecannotgiveasimplealgorithmthatwillalwaysleadtoasolutionbyinspectioninamanageableway.Readersmaywishtoconsiderafewrulesofthumb(illustratedwithexamples)inordertostartbecomingfamiliarwithsuchasolving.RULE1.Weinspectonebyonealltheequationsin(3.3)-(3.4),startingwiththosewhoseleft-handside(dωi)isnotzero.Thepolarmetricinξ2:ds2=dρ2+ρ2dθ2.Wearegiventhismetricandtheconditionofzerotorsionandweareaskedtocomputetheωi’sandων’s.iWereadilyobtain1212ω=dρ,ω=ρdθ,ω2=−ω1.(6.1)Thefirstequationsofstructurebecomedω1=0=ω2∧ω1,(6.2)2dω2=dρ∧dθ=ω1∧ω2.(6.3)1Becauseofrule1,wefirstinspect(6.3),whichleadsustoprovisionallyassumethatω2isproportionaltodθ.Wenowinsertω1=−dθin(6.3)andfindthat,12indeed,itissatisfied.Soω2=−ω1=dθisthesolution.12ItmayappearthatEq.(6.3)wasnotusedforfindingthesolutionandthat,therefore,thereisredundancyintheequations.Thisisnotcorrect.ThemostgeneralsolutionofEq.(6.3)isω2=dθ+fdρ,wherefisanarbitraryfunction1ofρ.Ofallthesesolutions,Eq.(6.3)pickstheonewithf=0.However,whenwesolvebyinspection,wefirstfocussolelyonthetermdθthatmustbepresentinω2forEq.(6.3)tobesatisfied,notthosewhich,likefdρ,mightormight1notbepresent.Wethusformulatetherulethatfollows.RULE2.Inordertoavoidasmuchaspossiblethecarryingofunknowncoefficients,weonlyextractfromeachequationtheinformationthatsomeωνimustcontainatermproportionaltosomespecificωλ.Wedonotconsidertheothertermswhoseexistencemaynotbeneededuntillateron.Ateachstage,everythingthatisnotnecessarilydifferentfromzeroisprovisionallyzero.Thepolarmetricin2-DLorentzspace.Considerthemetricdρ2−ρ2dθ2andzerotorsion.Wenowhave:ω0=dρω1=ρdθω1=ω0.(6.4)01Fromdω1=dρ∧dθ=ω0∧ω1,wefindthat,provisionally,ω1=dθ.Since00ω0=dθsatisfiesthedω0equationofstructure,weconcludeω1=ω0=dθ.101

200182CHAPTER9.EUCLIDEANCONNECTIONSThesphericalmetricinξ3:ds2=dr2+r2dθ2+r2sinθdφ2.Wereadilywriteω1=dr,ω2=rdθ,ω3=rsinθdφ,ωj=−ωi.(6.5)ijHence,dω2=dr∧dθ=r−1ω1∧ω2=ω1∧ω2+ω3∧ω2(6.6)133−1131322dω=sinθdr∧dθ=rω∧ω=ω∧ω1+ω∧ω2(6.7)dω1=0=ω2∧ω1+ω3∧ω1.(6.8)23From(6.6),ω2=r−1ω2(provisionally).Ifω2containsatermproportional11toω1,andω2containsatermproportionaltoω3,thesetermswillshowup3wheninspectingotherequations.Forthesamereason,ifω2containsaterm1proportionaltoω3andω2containsatermproportionaltoω1suchthattheir3respectivecontributionscancelwitheachother,wedonotbotherforthetimebeing.Wenowexamine(6.7).Necessarily,ω3mustcontainthetermr−1ω3.1So,provisionally,ω3=r−1ω3.Theseprovisionalων’sbecomethesolutionsince1itheyalsosatisfythelastequationtobechecked,(6.8).RULE3.Ifweareabletomatchbyinspectionalltheequationswithprovisionalων’s,theybecomethesolutionduetotheuniquenessthatwasprovedμinSection3.TheSchwarzschildmetricinisotropiccoordinates.Itisgivenbyds2=(ω0)2−(ωi)2,(6.9)whereω0=eφ(r)dt,ω1=eμ(r)dr,ω2=eν(r)rdθ,ω3=eλ(r)rsinθdφ.(6.10)Weproceedtoinspecttheequationfordω0,usingprimestodenotederivativeswithrespecttor:dω0=φe(φ−μ)ω1∧dt=...(6.11)Wedonotneedtowriteoutωρ∧ω0.Weprovisionallyhaveρω0=φe(φ−μ)dt.(6.12)1Weskipforthemomentthedω1equationandconsidertheequationfordω2:dω2=(1+μr)eμdr∧dθ=(1+μr)ω1∧dθ=ωλ∧ω2.(6.13)λdθisproportionaltoω2,whichcannotbethefirstfactorinωλ∧ω2sinceω2isλ2zero.Wecomparedr∧dθwithω1∧ω2andobtain,provisionally,1ω2=(1+μr)dθ=−ω1.(6.14)12

2019.6.LEVI-CIVITACONNECTIONBYINSPECTION183Thiswasthefirstappearanceoftheω2(equivalentlyω1)forms.Wenowinspect12dω3,wherewewriteallthedxμintermsoftheωρ’sexceptfordφ.Thus312dω=sinθ(μr+1)ω∧dφ+cosθω∧dφ.(6.15)Thisrequiresthat,necessarily,ω3andω3containatleastthetermsexhibited12in:ω3=sinθ(rμ+1)=−ω1,(6.16)13ω3=cosθdφ=−ω2.(6.17)23Wehadnotdetectedanon-nullω3termindω2.Wethusreturntodω2:2dω2=...ω3∧ω2.(6.18)3In(6.17),ω2isproportionaltoω3sothatω3∧ω2iszeroandtheequationfor33dω2continuestobebalanced.Wenowinspectthedω1equation.Itinvolvestheω1termswithρ=1.Theprovisionalresults(6.12)-(6.14)satisfytheequationρdω1=0=ωρ∧ω1.(6.19)ρOnefinallycheckseverything,justincase.RULE4.Consideratermforωνemergingintheequationfordων,andμwhosebeingdifferentfromzerowasnotapparentwhenpreviouslyconsideringtheequationfordωμ.Onereinspectsthedωμequationtoseeifsaidnewterm—alsopresentinωμ—yieldsornotacontributiontodωμ.Ifitdoesnot,weνkeepinspectingnewequations.Ifitdoes,seethenextexample.ThevanStockum’smetric.Itisdefinedby2222ω0=dt−ar2dφ,ω1=e−αr/2dr,ω2=e−αr/2dz,ω3=rdφ,(6.20)withsignature(+1,-1,-1,-1),whereαandaareconstants.Forpresentpurposes,weshallseta=α=1.Theequationfordω0reads2dω0=−2rdr∧dφ=−2er/2ω1∧ω3=ω1∧ω0+ω2∧ω0+ω3∧ω0.(6.21)123Theexteriorproductω1∧ω3mayarisefromeitherω1∧ω0orω3∧ω0,orboth13atthesametime.Soweleaveitforfutureexamination.Weproceedtoexaminetheequationfordω2:222−r/2r/212μ2dω=−redr∧dz=−reω∧ω=ω∧ωμ.(6.22)μSinceω2iszero,ω1∧ω2canonlyarisefromω1∧ω2.Thus,weprovisionally21have2ω2=−rer/2ω2=−ω1.(6.23)12

202184CHAPTER9.EUCLIDEANCONNECTIONSFromtheequationfordω3,weobtain23r/2−113μ3dω=dr∧dφ=erω∧ω=ω∧ωμ,(6.24)μwhichprovisionallyyields2ω3=er/2r−1ω3=−ω1.(6.25)13Wethengotothedω1equation.Wesubstitute(6.23)and(6.25)indω1=0=ω0∧ω1+ω2∧ω1+ω3∧ω1,(6.26)023andobtaindω1=0=ω0∧ω1,(6.27)0whichimpliesthatω1isproportionaltoω0andhasnoothercomponents.0Wehavetoreturntotheequationfordω0,whoseconsiderationhadbeenpostponed.Ifω1isproportionaltoω0,asrequiredby(6.27),itisnotpossible0tomatchω1∧ω0withanythingelsein(6.21).So,weenteranewphaseinthe1computation.Becauseitappearsthatwehaveaproblemwithω0,wereturntothedω01equation,(6.21),andassumethatthetermω1∧ω0willmakepartofmatching1thelefthandside.Thismeansthatω0willhaveatermwhereω3isafactor.1Thisinturnleadsustoconsiderthatω0mayhaveatermwhereω1isafactor.3Wethussetω0=Aω3,ω0=Bω1.(6.28)13Substitutionintheequation(6.21)fordω0yields−2er/2=A−B.(6.29)Becauseω1=ω0,wemustrechecktheequationfordω1,namely(6.26):010=ω0∧Aω3+ω2∧ω1+ω3∧ω1,(6.30)23whichindicatesthatω1mustnecessarilycontainatermproportionaltoω0,in3additiontothetermexhibitedin(6.25):2ω1=Aω0−er/2r−1ω3.(6.31)3Inthisnewphase,westillhavetoconsiderthedω2anddω3equations.Wethusreturnto(6.24).Bothω0andω1entertheequationfordω3:3322dω3=er/2r−1ω1∧ω3=−Bω0∧ω1−ω1∧Aω0+ω1∧er/2r−1ω3+ω2∧ω3.2(6.32)Thematchingoftheω0∧ω1termsrequiresA=B.Hence,using(6.29),2A=B=−er/2.(6.33)

2039.7.STATIONARYCURVESANDEUCLIDEANAUTOPARALLELS185Thus,tentatively,221r/20r/2−133ω3=−eω+erω=−ω1,(6.34)2ω1=−er/2ω3=ω0,(6.35)012ω0=−er/2ω1=ω3.(6.36)30Noticethat,inprinciple,ω3neednotbezero,sinceitcouldbeproportionalto2ω2.Butitisnotnecessarilydifferentfromzeroatthispoint.Oneeasilyverifiesthatalltheequationsforthedωμareverifiedwiththosevaluestogetherwith0=ω0=ω2=ω2=ω2,(6.37)2033whichthusyieldsthesolutionFromtheexamplejustsolved,weextractafinalrule:RULE5.ifwecannotmatchthecoefficients,weintroduceundeterminedcoefficients(AandBintheexample).Intheunlikelyeventthattheprocessbecomestoomessy,resorttocoordinatebasesanduseformula(4.5).Youcanalwaysreturntotheorthonormalbasesknowinghowtheconnectiontransforms.9.7StationarycurvesandEuclideanAUTOPARALLELSAdifferentiablemanifoldendowedwithanEuclideanconnectionhasametricstructuresuperimposedonanaffinestructure.Theaffinestructureallowsforaconceptofautoparallels,orlinesofconstantdirection.Themetricstructuregivesrisetoaconceptofstationarylengthbetweentwopoints,notnecessarilymaximumorminimumlength.Wesawinsection8.7howtoturnintoequationstheconceptofautoparal-lels.Themetricstructurenowprovidesuswiththelengthofcurvesasamostdesirableparameter,sincetheautoparallelsthensatisfytheequationdt=0,asfollowsfromdifferentiationwithrespecttosofdPdPds2t·t=·==1,(7.1)dsdsds2sincedt/dsisperpendiculartotandtheirdotproductiszero.Whenweusesasaparameter,wechoosetorefertotasu.Wethenhaveduν+uμωννduμduμνλ0==eν=+uΓμλueν,(7.2)dsdsdsμμλωλwherewehavereplacedωνwithΓνλω,andwheredsmeanspreciselythefactoruλofdsintheexpressionforωλ.Needlesstosaythat,sincedP=ωλe,theλ

204186CHAPTER9.EUCLIDEANCONNECTIONSuλarecomponentsofthe4-velocity(inthecaseofspacetime).Equation(7.2)appliesinparticulartocoordinatebasisfields,i.e.x¨ν+Γνx˙μx˙λ=0,(7.3)μλwhereoverdotmeansdifferentiationwithrespecttodistance,s.Itthusmeansdifferentiationwithrespecttopropertime,ifdealingwiththespacetimemani-fold.Ofcourse,theissuebecomeswhetherautoparallelswouldmeananythingifthetorsionofspacetimewerenotzero.Considernowcurvesofstationarydistance.Bydefinition,wehavePPμνdxdx0=δds=δgμνds,(7.4)QQdsdswhereδmeansvariationandwhere(P,Q)arethecommonendpointsofafamilyofcurves.Werewrite(7.4)asP0=δLds,(7.5)QwhereL=gx˙μx˙λ.(7.6)μλTheEuler-Lagrangeequations,d∂L∂L=,(7.7)ds∂x˙ρ∂xρfollow.Straightforwarddevelopmentof(7.7)yieldsλμλ1μλgσλx¨+gσλ,μx˙x˙−gμσ,λx˙x˙=0.(7.8)2Contractionwithgνσandaminormanipulationresultsinν1νπμλx¨+g(gμπ,λ+gπλ,μ−gμλ,π)˙xx˙=0.(7.9)2Weuseeq.(4.5)tofurtherwrite(7.9)asx¨ν+Γνx˙μx˙λ=0,(7.10)μλwhichistheequationfortheextremals.Comparisonwith(7.3)showsthatthisequationisthesameastheequationoftheautoparallelswithαasconnection.ThissaysthattheextremalscoincidewiththeautoparallelswhentheEuclideanconnectionistheLCC,i.e.inRiemanniangeometry.Beforeweproceedwiththerelationbetweenautoparallelsandextremalsinthegeneralcaseofnon-zerotorsion,onemustbeawareofthefollowing.Ingeneralrelativity,a“particle”havingamasscomparabletothemassofnearbybodiescannotbeviewedasatestparticle.Itwillfollowgeodesicsdetermined

2059.7.STATIONARYCURVESANDEUCLIDEANAUTOPARALLELS187bythemetricthattheparticleitselfhelpstocreate,byvirtueofitsaffectinghowtheotherparticlesmove.Henceithasanindirecteffectonitsownmotion.Inahypotheticalworldwithtorsion,autoparallelsandgeodesicswouldnotcoincideingeneralsinceωμ=Γνωλ=Γνωλ+Lνωλ,(7.11)νμλμλμλwheretheLν’sarethecomponentsofthecontorsion.Ifthegravitationalequa-μλtionsofmotionwerestillgivenbygeodesics,thegravitationalinteractionwouldνnotbedirectlysensitivetothetorsion,sinceΓμλisindependentofit.Wesaid“wouldnotbedirectlysensitive”becausethetorsionmightstillcontributetohowchargedparticlesinteractthusaffectingthegravitationalfieldtheycreate.Aninterestingissueis:whattypeofcontributiontotheequationsofmotionwouldthetorsionyieldiftheautoparallelsweretheequationsofmotion?ForthatweneedonlyconsiderLνωλ,sincethecontributionofΓνωλisadditivetoμλμλitandhasalreadybeendiscussedandassignedtogravitation.Thequestionthatweareaskingistoogeneral.WeshallaskthesimplerquestionofcontributionoftorsionsofthespecificformΩμ=−vμF,(7.12)wherevμissomevectorfieldandFisascalar-valueddifferential2−form.Inviewoftheconsiderationsmadeabouttheadditivityoftheeffectsofthemetricandthetorsion,wemayassumethatthemetricisflatforpresentpurposes.Wethenhave0=¨xν+Lνx˙μx˙λ.(7.13)μλWeraisetheindexνinEq.(5.8):ν1νννLμλ=(Rμλ+Rλμ+Rλμ).(7.14)2Thecontractionwith˙xμx˙λannulsthemiddletermbecauseofskew-symmetryofRνwithrespecttothesameindices.Wethenusehere(7.12)andobtainλμν1ννμλ0=¨x−(vμFλ+vλFμ)˙xx˙,(7.15)2i.e.νμνλx¨=+(vμx˙)Fλx˙.(7.16)Ifonecouldreplacevwith˙x(inwhichcasevx˙μwouldbetheunity),Eq.μμμ(7.16)wouldrepresenttheequationofmotionofspecialrelativitywithLorentzforce,providedthatoneweretothinkofFastheelectromagnetic2−formuptoanappropriateconstant.Inotherwords,theelectromagneticfieldwouldbecomeassociatedwiththetorsionofspacetime[72],[64].AtorsionFx˙λactuallyisFinslerian.IntheFinslerbundle,the4-velocityνλisnotafieldoncurvesorredundantcoordinatesinthefibers.Thesecoordi-natesnowbelongtotheFinslerianbasemanifold,andtheybecomevelocity

206188CHAPTER9.EUCLIDEANCONNECTIONScomponentsoncurvesofthatmanifoldthatarenaturalliftingsofspacetimecurves.TheproperwaytowritethetorsionFx˙λintheFinslerbundleisνλΩ0=−F.SincethegroupinthefibersofFinslerianspacetimeistherotationgroupinthreedimensions,Ω0isaninvariant;itdoesnotacquireΩitermsinotherframefields.FinsleriantorsionshavetermsthatlookFinslerianandothersthatdonot(butstillare!).Whenthefirstonesarenull,theequationoftheautoparallelsdependsonΩ0butnotonΩi.IttakestheformofequationofmotionwithLorentzforce(themetricwouldstillcontributewiththegravitationalforce)[81].9.8EuclideanandRiemanniancurvaturesTheEuclideanandRiemanniancurvaturescoexistandarerelatedthroughtheνcontorsion,βμ,asfollows:dω−ω∧ω≡dων−ωλ∧ων=d(αν+βν)−(αλ+βλ)∧(αν+βν).(9.1)μμλμμμμλλThisequationwouldtakeafarmorecomplicatedformifweexpressedthecontorsionintermsofthetorsion.Wereorganize(9.1)toreaddω−ω∧ω=dα−α∧α−β∧β+(dβ−α∧β−β∧α).(9.2)Thenotationintroducedshouldbetakenintoaccountinordernottoinadver-tentlysetα∧α,β∧βetc.equaltozero.TheseequationsmayberewrittenasΩν=Ων−(β∧β)ν+(dβ)ν,(9.3)μμμαμwith(dβ)νascomponentsrelativetoeμ∧eoftheexterior(-covariant)deriva-αμνtiveofβwhencomputedasifανweretheconnection.Noticethat(β∧β)νμμλνsimplymeansβμ∧βλ.Substitutionofω−βforαinthelastparenthesisof(9.2)and(9.3)yieldsdω−ω∧ω=dα−α∧α+β∧β+(dβ−ω∧β−β∧ω),(9.4)andΩν=Ων+(β∧β)ν+(dβ)ν,(9.5)μμμωμrespectively.The(dβ)νnowarethecomponentsrelativetoeμ∧eoftheωμνexterior(-covariant)derivativeofβwithrespecttotheactualEuclideancon-nectionofthedifferentiablemanifold.Inthetensorcalculus,relationsofthisnaturetakeverycomplicatedforms.Inordertocontinuedissuadingreadersfromthinkingthattheywouldhaveanadvantageinworkingwithtensorsindealingwithequationssuchastheprecedingones,wereproduceanequationofhistoricalinterest,sentbyCartantoEinstein[29].Cartanwrote:

2079.8.EUCLIDEANANDRIEMANNIANCURVATURES189γ“WouldyouliketoknowhowtoexpressRαβintermsofΛαβ?Onehasβαρβρα2Rαβ=Λαμ;μ+Λβμ;μ−φα;β−φβ;α+ΛαμΛμρ+ΛβμΛμρ+SαSββασσ−(Λαρ+Λβρ)φρ+2ΛαρΛβρ−gαβSμSμ.”(9.6)ThisRαβisourRαβ,i.e.fortheLCC.ThesymbolsusedhadbeendefinedbyEinsteininapaperonteleparallelism[39].Underlinedindicesmeanthattheyhavebeenraisedorloweredinthestandardwaythroughthemetric(onlythenotationisnotstandard).Underlinedsubscriptsmust,therefore,becon-sideredasraised,thusdenotingcontravariantcomponents(Einstein’swayofdoingthingsinthisregardletsonerememberinwhatpositionagivenindexinaparticularquantitywasborn).φisΛα,andΛisgivenasμμααααΛμν≡Δμν−Δνμ,(9.7)withΔintroducedasδAμ≡ΔμAαδxβ,(9.8)αβwhichisthereplacementforδAμ=−ΓμAαδxβ(9.9)αββinvolvingtheLCC.Withtheindicesα,β,γandδbeingalldifferent,SαisSγδ,whichinturnisdefinedasSα≡Λα+Λν+Λμ.(9.10)μνμναμναWewishtowarninexperiencedreadersaboutmisusingtheannulmentoftheconnectionatapoint.Itimpliesβ=−α,butnottheequalityofdβand−dα.Thatwouldrequiretheannulmentoftheconnectionalsoonaneighborhoodofthepoint.Replacementofβ=−αindβwouldbeincorrect.In(9.2),itwouldforcetheEuclideancurvaturetobezeroatthatpoint,whichmightbeincontradictionwithwhateverassumptionsmighthavebeenmadeabouttheEuclideancurvatureinthefirstplace.

208190RestrictionAFFINECONNECTIONS−−−−−−−→EUCLIDEANCONNECTIONSofbundleeμ·eν=δμνωλμ,ωνωμν+ωνμ=0d¯P∗=ωμeμdωμ=ων∧αμαννd¯e∗=ωνeμμμναμν+ανμ=0CHAPTER9.EUCLIDEANCONNECTIONSνννβμ≡ωμ−αμ,βμν+βνμ=0d˙d¯P=Ωμ∗eμΩμ=dωμ−ων∧ωμνμνμν∗νλνd˙d¯eν∗eνΩν=dων−ωλ∧ωνΩ=−ω∧βνΩμ=dαμ−αμ∧αλμ=ΩμμμμλdΩμ=−Ων∧ωμ+ων∧ΩμdΩν=−Ωλ∧ων+ωλ∧ΩνννμμλμλdΩν=−Ωλ∧ων+ωλ∧ΩνμμλμλννΩ=Ω−β∧β+(dβ−α∧β−β∧α)orμμννΩμ=Ωμ+β∧β+(dβ−ω∧β−β∧ω)∗d¯isnotanoperatorbutfirsthalfofsymbols¯dPand¯de.d˙:formaldifferentiation;d˙d˙=0.Ωμ≡Rμ(ων∧ωλ),Ων≡Rν(ωλ∧ωπ),Ων≡Rν(ωλ∧ωπ).νλμμλπμμλπEuclideanconnectionssatisfyformulasontherightboxinadditiontotheformulasontheleftbox.

209Chapter10RIEMANNIANPSEUDO-SPACES&RIEMANNIANSPACES10.1KleingeometriesingreaterDETAILAKleingeometryisageometryinthespiritoftheErlangenprograminitsmoderninterpretation.Ifageometryisnotofthattypeoranappropriategeneralizationthereof,themanifoldinwhichitliveswouldbereferredbyCartanasapseudo-space.HeactuallyreferredtoRiemannianmanifoldsinthefirstdecadesoftheirlivesaspseudo-spaces.Moreonthesepseudo-spacestocomeinthisandthenextsection.AKleingeometryisnowadaysdefinedasapair(G,G0)ofagroupGandasubgroupG0havingacertainimportanttechnicalproperty.Thereisnotunanimityastowhatthispropertyshouldbe(Fordetailssee[27],[67],[82]).Webelieve[27]tobetherightone.CartanspokeofKleingeometriesasthestudyofinvariantsunderthetrans-formationsofagroup.Butitisclearfromhiswriting,thoughimplicitly,thathehadinmindpairsofagroupandasubgroup.Hedidnotneedtoelaborateontheconceptsince,forthemostpart,hestudiedgeneralizationsofspecificKleingeometries.ThegroupsGofthosegeometrieshavesubgroupsG0withtherequiredproperty.G0happenstobethesubgroupconstitutedbyalltheelementsofGthatleaveapointunchanged.ThetransformationsinG0arepresentinthefibersofthegeneralizedspaces.Thisisincontrastwithtransformationsliketranslations,whicharevalidonlyindifferentialforminthegeneralizations(seethePreface).LaterinhislifeCartanwroteexplicitlyandatlengthabouttheroleofgrouptheoryinmoderndifferentialgeometry[21].Inhispresentation,hementionedG0repeatedly,butonlyexceptionallydidherefertothepairofgroupandsubgroup.191

210192CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESInEuclideangeometry,Gisthegroupofthesocalleddisplacements(trans-lationsandrotations,withthegroupofrotationsasG0).InaffinegeometryGisthegroupoftheaffinetransformations(translationsandlineartransfor-mations,withthelineartransformationsasG0).Inprojectivegeometry,Gistheprojectivegroup(translationsandhomographies,withthehomographiesasG0).Again,thetransformationsinG0arethetransformationsfromGthatleaveapointunchanged.TheGgroupiscalledthefundamentalgroupofthecorrespondinggeometry.Onealsospeaksofthefundamentalgroupofthespacewherethestudyofthepropertiesofthefigurestakesplace.WithCartan’sownwords[15]:Ineachofthesegeometriesandforexpediencyreasons,oneattributestothespacewherethefiguresunderstudyarelocatedtheverypropertiesofthecorrespondinggroup,orfundamentalgroup;onethusspeaksof“Euclideanspace”,“affinespace”,etc.,insteadof“spacewhereonestud-iesthepropertiesofthefiguresthatareleftinvariantbytheEuclideangroup,affinegroup,etc.”Weproceedtoexplainwhat,inthemodernview(readCartan’s),waswronginKleinhimself’sviewofhisErlangenprogram.In[21],Cartanstatedthatananalytictransformationfromxitoxitransformsthedifferentialformgdxidxjijintogdxidxj.TheformulasthatshowhowtogofromthexiandthegtotheijijxiandgdefineaninfiniteLiegroupdependingonn+[n(n+1)/2]variables.ijThisisthegroupthatKleinconsideredasunderlyingRiemann’sgeometry.WeshallskipfurtherdetailsofthisKleineanview(“inappropriate”accordingtoCartan)ofRiemanniangeometryandofitsevolutionintheworkofVeblenandoftheschoolofHamburg(perCartanreport[21]).Afterall,weareonlyinterestedinCartan’sviewofhowRiemanniangeometryfitshisgeneralizationofwhatshouldhavebeenKlein’sviewofgeometry.Cartanclaimedthat“thedevelopmentofcontemporarygeometry”(again,actuallymeaninghisowndevelopments)alsospringsfromRiemanniangeom-etry,butinthestageofdevelopmentachievedbythisgeometryafterthedis-coveryofthenotionofparallelism[21](Earlierinthepaper,CartanhadgivencredittoLevi-CivitaandSchoutenfortheiralmostsimultaneousdiscoveryofthisnotion).Cartanfurtherclaimedthat,inhiswork,thenotionofgroupintervenesinafarmoreprofoundwaythanithavebeenthecase.ItcontinuestobethecasewiththosewhomaintainthatthegroupthatdefinesRiemanniangeometryistheinfinitegroupofdiffeomorphisms.HeexplainedthatthisgeometryinvolvesnotionsofaEuclideannature,suchasdistance,angle,area,volume,etc.Hefurtherstatedthat,althoughthenotionofgroupwouldseemtobeabsentfromthisgeometry,itis,however,presentindirectly,sinceallthesegeometricnotionshavetheirorigininthegroupofEuclideandisplacements.Infact,theframesconstitutedbyvectorswithoriginatapointAofaRiemannianspace(after1917)arerelatedbythegroupofrotationsaroundA,andthetheoremsrelativetothesefiguresarethesameasinEuclideangeometry.Inotherwords,the

21110.2.THEFALSESPACESOFRIEMANN193rotationsofframes(i.e.theelementsofG0)surviveinfiniteforminRiemanniangeometry.Thisistobecontrastedwiththetranslations.StillreferringtogeneralizationssuchasRiemanniangeometry(again,after1917),Cartanstatedthat“thenotionofparallelismpermitsequallytogiveacertainsensetotheinfinitesimaltranslations”.Thekeywordhereisinfinites-imal,whichwasnotrequiredinthecaseofrotations(wedonotadvocatetheuseoftheterminfinitesimalasfreelyasCartandidalmostacenturyago,buttherearecases,likethepresentone,whereithelpstoavoidclutter.)ThegroupGalsoisthesamegroupfortheKleingeometriesasfortheirrespectivegen-eralizations,butonlyindifferentialform(Liealgebra)whentheelementsofGthatarenotinG0areconcerned.10.2ThefalsespacesofRiemannThissectiondealswithwhatCartancalled“falsespacesofRiemann”,termnottobeconfusedwithpseudo-Riemannianspaces,oftenusedwhenthemetricisnotpositivedefinite(Laterinthechapter,thetermRiemannianspacewillbeanotherwaytorefertodifferentiablemanifoldsendowedwithazero-torsionEuclideanconnection).Cartanwrote[13]:”Withhisdefinitionofparallelism(1),Levi-Civitawasthefirsttosuc-ceedinmakingthefalsemetricspacesofRiemann,ifnottrueEuclideanspaces,whichisimpossible,atleastspaceswithaEuclideanconnection,consideredascollectionsofsmallpiecesofEuclideanspace...”Emphasisisasintheoriginal,wherethetermsfalseandspacescometogetherin“fauxespacesmetriquesdeRiemann.”Thereference(1)istothefamousLevi-Civitapaperof1917[52].Attheendofthe19thcentury,thereweretwomaindevelopmentsinge-ometry.OnewasduetoRiemann[61],[63]andtheotheronetoFelixKlein[49].Riemann’swasbasedontheconceptofdistance.Klein’swasbasedontheconceptofgroupand,relatedtoit,ontheconceptofgeometricequality.WeproceedtobrieflycitefromCartan(ibid)ongroupsandgeometricequality[13]:“Thenotionofgroup...becameingeometry,thankstoF.KleinandS.Lie,aclasificationprinciplewhichcompletedtheremovalofclassicalEuclideangeometryfromtheprivilegedpositionthatithadoccupiedforsuchalongtime;thisgeometryisinfactnothingbutthestudyofthosepropertiesoffiguresthatareconservedbyacertaingroupoftransforma-tions(thedisplacements),andtheaxiomsofgeometricequalitysimplyexpressthepropertyofthesetransformationsofconstitutingagroup,asmadeevidentbyPoincare”.CartangeneralizedtheKleingeometries,i.e.thegeometriesofflataffine,Euclidean,conformal,projective,etc.spaces.Hence,thisisnotthesamesenseofgeneralizationwhereaflatgeometrygeneralizesanotherflatgeometry(likeaffinegeometrygeneralizesEuclideangeometry).

212194CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESThefalse(i.e.theold)RiemanniangeometrydidnotgeneralizetheKleineanEuclideangeometry.ItgeneralizedthemetricpartbutnottheaffinepartoftheEuclideanstructurebecauseitdidnothaveanyaffinecontent.Itisonlyafter1917thatonecansaythatRiemannianspacesmaybeviewed“asif”madeofsmallpiecesofEuclideanspace[15].Wesaid“asif”because,forexample,verysmallpiecesofathinglassspheremaylookflat,buttheradiusofthesphereremains“written”ineverypiece,regardlessofsize.InhispaperontheroleofLiegrouptheoryintheevolutionofmoderngeometry,Cartan[21]emphasizedthedifferenceinviewpointbetweenKleinandRiemannasfollows:“...whereasRiemanniangeometryisasimplegeneralizationofEu-clideangeometry,Kleinretainsfromthelatteraboveallthenotionofgeometricequality,Riemannretainsonlythenotionofdistance.Pushedtotheirlastconsequences,thetwoviewpointsareradicallydivergent:thenotionofdistancedisappearsfromthemoregeneralKleingeometriesandthenotionofequalfiguresdisappearsfromthemoregeneralRie-manniangeometries;thenotionofgroupceasestobeatthefoundationoftheRiemanniangeometries...”Tobemorespecific,thequantitiesthatweknowascomponentsofthedif-ferentialformανwerealreadyusedintheoldRiemanniangeometry,buttheyμdidnotplaythenanyaffinerole,andtheconceptofdifferentialformsdidnotyetexist.InfactthegeneraltheoryofaffineandEuclidean(andconformal,andprojective,etc.)connectionswasnotformulateduntiltheearly1920’sbyCartan(Seethereferenceswehavegivenofhisworkofthoseyears).Onemustsay,however,thataffineconsiderationswereimplicitinRicciandLevi-Civita’smonographof1901[60],andspecificallyintheirderivationoftheformulaforthecurvatureinRiemannianspaces.Theyare,however,disguisedasasearchfortensorialquantites.Thisprimitively-affineapproachtothecurvaturewasreproducedbyEinsteininhisworkongeneralrelativity[37].Inthesamepaper,Cartan[21]characterizedtheoldRiemanniangeometryas“thetheoryoftheinvariantsofaquadraticdifferentialform,gdxidxν,oniνnvariablesxiwithrespecttotheinfinitegroupofanalytictransformationsperformedonthesevariables”.Theobtainingofthoseinvariantsisthecoreoftheproblemofdeterminingwhethertwogeometricobjectsareequivalent,i.e.relatedbyacoordinatetransformation.Hencethisisoneoftheso-calledproblemsofequivalence,whichRiemannsolved[62].AsreportedbyPauliinhisbookonthetheoryofrelativity[57],ChristoffelandLipshitzalsosolvedthesameproblemofequivalence,independentlyofeachotherandofRiemann.Inthenextsectionweshallprovideamoredetailedsummarythaninsection7.5ofCartan’sderivationoftheRiemanniancurvatureasaninvariantofthemetric,usingagainthemethodofequivalence.Whatmakesitaninvariantisthatitisadifferentialforminabundle.WeareusingthetermbundlestilllessspecificallythanwehavedonesofarinordertoreflecttheveryinformalwayinwhichCartanintroducedit,asweareabouttosee.

21310.3.METHODOFEQUIVALENCE19510.3Riemannianpseudo-geometryandmethodofequivalenceRiemann’spaperofgreatestgeometricrelevanceafterhisseminalpaperonthefoundationsofgeometryistheoneinwhich,forthefirsttimeinknownmathematicalhistory,aformulaforthecurvatureofamanifoldofdimensionthreeorgreaterthanthreewasderived[62],[63].Butitwasnotpreciselyageometricpaper,inspiteofthatresult.Itwasapaperonheattransfer,writtentorespondtoaproblemposedbytheFrenchAcademy,knownasParisianAcademyatthetime.Thecurvatureappearednotasageometricconcept,butasasetofquantitieswhosebeingnullsatisfiedtheproblemofequivalencethatemergesinthetheoryoffalsespacesofRiemann.Itdidnotlookmuchasgeometry.ThebodyofpropositionsofRiemanniangeometrywasstillinitsinfancywhenRiemanndiedin1866.WenowreproducethemainlinesofCartan’sderivation[8]oftheRieman-niancurvatureandrelatedequationsasanapplicationofhistechniquetosolveequivalenceproblems.Grosslyspeaking,Cartan’smethodofequivalencecon-sistsindifferentiatingasetofdifferential1−formsinabundlerepresentativeofaproblem,thendefiningnewquantitiesemergingfromthatdifferentiationandfurtherdifferentiatinganddefiningnewconceptsuntilonerunsoutofthingstodifferentiate.Forextensivetreatmentofthemethodofequivalencesee[42].Thedifferential1−formsrepresentativeofmetricsarethemostgeneralωμ’sthatdiagonalizeit:ds2≡g(x)dxμdxν=ε[ωλ(x,u)]2(3.1)μνλλwhereελ=±1.In2-DEuclideanspace,wehave12ω=cosαdx−sinαdy,ω=sinαdx+cosαdy.(3.2)Thereisonlyone“ucoordinate”inthiscase,namelyα.FortheLorentziansignature,theucoordinateisthecoordinateofthecorrespondingLorentzgroup.Theexteriordifferentialsoftheωμ’scanbewritteninaninfinitenumberofwaysasdωμ=ενων∧ανμ.(3.3)νEquation(3.3)helpsdefinedifferentialformsανμ.Itwillberecognizedasthefirstequationofstructurewhenthetorsioniszero,exceptthatthetorsionisnotaconceptneededtosolvetheproblemofequivalenceandtheα’sarenotaffineorEuclideanconnectionshere.Theds2doesnotdependeitheroncoordinatesuortheirdifferentials.Weshalldesignatebydxandduthedifferentiationwithrespecttocoordinatesxandcoordinatesurespectively.ωμisafunctionofcurvesthatdoesnotdependonthedu’s.Thexanduinsuperscriptedparenthesesrefertothetermsrespectivelyproportionaltodxandduindifferential1−forms.Evidently:ω(u)=0,d(ds2)=0.(3.4)μu

214196CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESThesecondofequations(3.4)canbeexpandedasεω(x)dω(x)=0,(3.5)μμuμμwherejuxtapositionmeanstensorproduct.Ontheotherhand,thefirstof(u)equations(3.4)impliesdxωμ=0and,therefore,dω(x)=dω(x)−dω(u).(3.6)uμuμxμ(x)Cartanfurthertransformsduωμ,whichhethensubstitutesin(3.5).Thisallowshimtoderive(u)(u)α+α=0(3.7)λμμλasanecessaryandsufficientconditionfor(3.3)tobesatisfied.Hethusprovesthattheskew-symmetricpartofα+αislinearinthedxμand,therefore,λμμλintheωμ.Cartanthenshowsthattheremainingfreedomtochoosetheα’scanbeused(u)tomakeαλμ+αμλequaltozero.Supposewehadasetαλμ’ssuchthatitsαλμpartsatisfied(3.7)andthus(3.3).Letαλμbedefinedasν=nαλμ=αλμ+νβλμνων,(3.8)ν=1whereβisarbitraryexceptforsatisfyingtheconditionβλμν=−βμλν.(3.9)αλμalsosatisfies(3.7)and(3.3).Thesumαλμ+αμλthenisα+α=α+α+(β+β)ων,(3.10)λμμλλμμλλμνμλνGiventhehorizontalityofαλμ+αμλ,onecouldusethisfacttosolvefortheβλμνthatsatisfiesthesystemconstitutedby(3.9)andtheequationsαλμ+αμλ+βλμν+βμλν=0.(3.11)Thissystemhasauniquesolution,whichisobtainedasperthesolvingofsimilarsystemsinthepreviouschapter.Wethushave,droppingthefrom-now-onunnecessaryunderliningofαλμ,αλμ+αμλ=0.(3.12)Itisclearthatjustanumberofthen(n−1)/2areindependent.Cartanproceedstodifferentiate(3.3)andtothensubstituteintheresulttheequations(3.3)themselves.Hethusobtainsωμ∧(dαν−αλ∧αν)=0.(3.13)μμλ

21510.4.RIEMANNIANSPACES197ThisleadshimtodefinethedifferentialformversionoftheRiemanniancurva-tureasdαν−αλ∧αν.(3.14)μμλInprinciple,Ωisalinearcombinationoftermsωρ∧ωλ,ωμ∧ασandαν∧απμννμλsinceneitherofthetermsontherighthandsideof(3.14)ishorizontal.Cartanshowsthatthenon-horizontalpartsofdανand−αλ∧ανcanceleachotherout.μμλAtthispoint,weabandonCartan’streatmentofthissubjectandconcentrateonhowhisargumentsolvestheequivalenceproblemofRiemann.Supposewearegivenasymmetricquadraticdifferentialform.IfitwereadisguisedformoftheCartesianmetric(i.e.reducibletoitbyacoordinatetransformation),wewouldcomputeitsανandthendαν−αλ∧αν.Ifthisμμμλcurvaturedifferentialformisnotzero,thegivenmetricisnottheCartesianmetric,sinceithastobezeroinanycoordinatesystemThisisaconsequenceofbeinghorizontal.Theargument(forthosefamiliarwiththetensorcalculus)isnotdifferentfromthefactthatifatensoriszerowithrespecttosomebasis,itiszerowithrespecttoanybasisThatisthenecessarycondition.Itisalsosufficient,butweshallnotinsistonthissincetheRiemanniancurvatureoftheoldRiemanniangeometryisgivenbythesamedifferentialformsastheEuclideancurvatureoftheLevi-Civitaconnection.Wehavealreadydealtfromtheperspectiveofthetheoryofthemovingframewiththesufficiencycondition.10.4RiemannianspacesAproperRiemannianspace(meaninghere“onewhichdoesnotqualifyasaRiemannianpseudo-spaceintheabovesenseoftheword”)isadifferentiablemanifoldendowedwithazero-torsionEuclideanconnection.Ifthetorsioniszero,soisthecontorsion.TheEuclideanconnection,ων,thencoincideswithitsμmetric-dependentpart,αν,andiscalledtheLevi-Civitaconnection(LCC).Itμthenfollowsthatthe(Riemannian)curvatureobtainedintheprocessofsolvingtheproblemofequivalenceofRiemannianmetricsisthesameasthecurvatureoftheLCC.AlltheequationsvalidforEuclideanconnectionsapplytotheLCCand,therefore,toRiemannianspaces.Andalltheequationsthatapplytoaffineconnectionswithzerotorsionalsoapplyhere.Thustheequationsofstructureare0=dων−ων∧αμ,(4.1)νΩν=dαν−αλ∧αν,(4.2)μμμλexceptthatwehavetoaddtothemthedistinctivemarkofEuclideanconnec-tions,namelyαμν+ανμ=0.(4.3)Theargument,ofcourse,isofadifferentnaturefromtheoneintheprevioussection,sincetheproblemofequivalencehasnotbeenposedinthemovingframeenvironment,whereitisonlyimplicit.

216198CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESTheaffineextensionof(4.3)isλλdgμν−αμgλν−ανgλμ=0.(4.4)ThecoincidenceofωνandανinthiscasemeansthatΓνequalsΓν.Hence,μμμλμλtheautoparallelsandtheextremalscoincideonRiemannianmanifolds,andthestationarycurves(extremalsinparticular)alsosatisfytheequationdu0=.(4.5)dsAsisthecaseforallEuclideanconnections,wehaveΩμν=−Ωνμ,(4.6)andthefirstBianchiidentitybecomesωμ∧Ων=0.(4.7)μFromννλπ1νλπΩμ≡Rμλπ(ω∧ω)=Rμλπω∧ω(4.8)2and(4.7),wereadilygettheannulmentofacyclicsumofcomponentsRν,μλπnamelyνννRμλπ+Rλπμ+Rπμλ=0,(4.9)whichiscalledthecyclicproperty,validwhen(4.7)issatisfied.TothesealgebraicpropertieswecanofcourseaddννRμλπ=−Rμπλ,(4.10)asforarbitraryaffineandEuclideanconnections.WealsorecallthepropertyRμνλπ=Rλπμν,(4.11)firstseeninsection2ofthepreviouschapterandvalidforEuclideanconnectionwhosetorsionhaszeroexterior(-covariant)derivative.ThecontractionsRνaredesignatedasR.Theybehavelikethecom-μλνμλponentsofatensor,whichisthereasonwhyonereferstoRμλasRiccitensor.Itissymmetric:R≡Rν=gνπR=gνπR=Rπ=R.(4.12)μλμλνμπλνλνμπλμπλμOnedefinesEinstein’stensorasonewhosecomponentsare1Gμλ=Rμν−gμνR,(4.13)2whereRisRμ.Onceagain,thisisnottheproperwaytolookeithertotheμEinsteinandRicciobjects,ortothesocalledenergy-momentumtensor.They

21710.5.ANNULMENTOFCONNECTIONATAPOINT199arenaturallyvector-valueddifferential3−forms,tobefurtherdiscussedinlatersections.Insection6ofchapter8,wesawthattheformulafortheaffinecurvaturedifferential2−formgaverisetoaconceptofandformulaforacurvaturetensor.Then,insection7,wesawtheroleofthistensorfordeterminingtheevolutionofthetangentvectortoafamilyofautoparallelsaswemovealongandacrossthecurvesinthefamily.Insection11ofthesamechapter,wespecializedthisresulttothezerotorsioncasetoobtain∂2n=tμRνtλnπe.(4.14)∂u2μλπνTheonlyEuclideanconnectionwithzerotorsionistheLevi-Civitaconnec-tion.HenceitistheonlyEuclideanconnectiontowhichEq.(4.14)applies.Wecanuseasparameteruthedistancesongeodesics,whicharealsoautoparallelsρinthiscase.Wethenhavetρ=dx≡x˙ρ,whichallowsustowrite(4.14)asds∂2n=Rνx˙μx˙ρnπe.(4.15)∂s2μρπνTheparameterλusedinthedefinitionofncouldbeany.Wecanchooseittobes,sothatnwillbeaunitvector.Thisequationisthencalledtheequationforgeodesicdeviation.Itdoesnotmeasuretheseparationofgeodesics;thestraightlinesissuedfromapointintheEuclideanplaneseparateeventhoughthecurvature(s)iszero.Formula(4.15)ratherspeaksofthe“acceleration”intheseparationofthegeodesics.ThereisaninfinitenumberofEuclideanconnectionsonthesamemetric.Thoughtheydifferbytheirtorsions,theyallhavethesamesetofgeodesics(again,curvesofstationarydistance).Equation(4.15)thusappliestothegeodesicsofalltheEuclideanconnections,butnottotheirautoparallelsun-less,ofcourse,wearedealingwiththeLevi-Civitaconnection.10.5NormalcoordinatesandannulmentofconnectionatapointNormalcoordinatesaretypicallyintroducedinRiemanniangeometrytoachievetheannulmentoftheconnectionatapoint,Levi-Civita’sinthiscase.Itisclearthat,forarbitraryEuclideanconnections,nocoordinatebasisfieldsoftangentvectorsmakesitzeroatapoint.Indeed,correspondingtocoordinateframefields,wehavedωμ=ddxμ=0.Iftheconnectionwerezeroatapoint,therighthandsideofthefirstequationofstructurewouldbezero.Inotherwords,thetorsionhadtobezeroatthatpointforannulmentoftheconnectionthroughnormalcoordinates.Whenthetorsionisnotzero,onestillcanachievesaidannulment,aswesawinchapter8,butnotintermsofcoordinatebasisfields.WeproceedtoobtainnormalcoordinatesbymeansofaprocedurethatCartanattributestoRiemann.Sinceitlookssomewhatadhoc,wefirstprovideanexampleofthegeometricideasthatunderlieit.

218200CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESConsidertwoperpendicularplanesthatcontainthediameterthroughthepointPoftangencyofaplaneπtoa2-sphere.Theirintersectionwithπdeter-minesanorthonormalframeatP(uptoorientation).CartesiancoordinatesonπcannowbeassignedtopointsonthesphereintheneighborhoodofPbywayofaperpendicularprojection.Inthiswaythemetricdx2+dy2ontheplaneisassignedtotheneighborhoodofPonthesphere.Itisanapproximation,thesmallertheneighborhoodthebetter.Considernowatangentorthonormalframe(ai)atapointPofadifferen-tiablemanifold.PointsMinanarbitrarilysmallneighborhoodofPdeterminegeodesicsgthroughP.LetubetheunittangentvectortogatP,directedtowardsM.Letcibetheproductsu·ai(Ifthemetricispositivedefinite,thesearethedirectionalcosines).ThenormalcoordinatesofParethequantitiesxμ=cμs,(5.1)wheresisthelengthofthearcfromPtoM.Saidbetter,thecoordinatefunctionsassigntoeachpointMthequantitiescμs.Thecμ’sareobtained,sureenough,byintegratingtheequationsforthegeodesicsofthegivenmetric,whichthusenterthepicture.Itisclearthat,onthegeodesicstangenttothevectorsa,wehavexμ=s.iLetusverifythatthecoordinates(5.1)aresuchthattheΓ’sbecomezeroatP.Wesubstitute(5.1)intothegeodesicequationd2xμdxνdxλμ+Γ=0,(5.2)ds2νλdsdsandobtainΓμcνcλ=0,(5.3)νλtobesatisfiedatPforallc’sinaneighborhood.Thearbitrarinessofthec’sμimpliesthattheΓarezeroatP.νλAsanapplicationofnormalcoordinates,weshowpropertyRμνλπ=RλπμνfortheRiemanniancurvature,originallyobtainedforarbitraryEuclideancon-nectioninsection2ofthepreviouschapter.Intermsofnormalcoordinates,wehave,atthepointwheretheconnectionbecomeszero,Ω=dα=(Γ−Γ)(dxπ∧dxλ)≡R(dxπ∧dxλ),(5.4)μνμνμνλ,πμνπ,λμνπλwithR=−RandR=−R.Usingequation(5.1)oftheμνλπνμλπμνλπμνπλpreviouschapter,namely,π1νπΓμλ=g(gμν,λ+gλν,μ−gμλ,ν),(5.5)2wefurtherget1Rμνλπ=(gνπ,μλ+gμλ,νπ−gμπ,νλ−gνλ,μπ)(5.6)2Symmetryofthetherighthandsideunderexchangeofpairs(μν)and(λπ)thenimplies(4.11).Thehorizontalityofthecurvatureinturnimpliesthatthisresultisvalidinanyframefield.

21910.6.EMERGENCEANDCONSERVATIONOFEINSTEIN’STENSOR20110.6EmergenceandconservationofEinstein’stensorContractionsofRiemann’stensorleadtothediscoveryofasecondranktensorwhosecovariantderivativeiszero.ItisnamedtheEinsteintensor.Itwasrelevantatthetime(butnowonlybecauseoftradition)sincethatpropertyoftensorswasassociatedwithconservationlaws,whichpertaintodifferentialforms,nottotensors.Inaffine-KleinandEuclidean-Klein(i.e.flat)spaces,conservationofvec-torortensor-valueddifferentialformsfollowstheannulmentoftheexterior(-covariant)derivatives.Thenon-scalarvaluednessposesproblemsunlessthereareconstantframefields(teleparallelism),sothattheintegrationcanthenbereducedtoscalar-valuedintegration.ThislimitationappliestoRiemanniange-ometryand,therefore,togeneralrelativityifweassumethatitsconnectionistheLCC(ItmayactuallyhappenthattheconnectionisteleparallelandonemistakenlybelievestobedealingwiththeLCC,asexplainedinsection8ofthepreviouschapter).LetusreporthowtheEinsteintensoremergesinthetensorcalculus.OnewritesexplicitlyintermsofChristoffelsymbolsthecovariantderivativewithrespecttotheLCCconnectionofRiemann’scurvature.Oneusesnormalco-ordinatesinordertomakethatequationmoremanageable.Onewritesdowntwomorecopiesofitobtainedbyperformingappropriatecyclicpermutationsofthreeofthefiveindices(See(6.1)).Therighthandsideofasuitablecom-binationofthethreeequationssoobtainediszero,andsomust,therefore,bethesamecombinationofthelefthandsides.OnethusobtainsνννRμλπ;ρ+Rμπρ;λ+Rμρλ;π=0.(6.1)Adoublecontractionofindices(ν=π,μ=λ)yieldsλ2Rρ;λ−R,ρ=0.(6.2)TheRiccitensorhasapropertywhichallowsonetorewritethisasλρ1λρλρ0=(R−gR);λ=G;λ,(6.3)2whichistheannouncedresult.Sinceallthedisplayedequationsaretensorial,wemaydisregardourhavingusednormalcoordinatestoobtainthisresult.Differentialformsallowonetosimplifytheaboveapproach,whileretainingitsflavor.Intheprocess,onecanmakeclearerhowtheseequationsarecon-nectedwiththeconservationofthecurvature,i.e.thesecondBianchiidentityforRiemann’scurvature.UndertheLCC,itcanbewrittenas0=d=Rνωρ∧ωλ∧ωπe∧eν.(6.4)μλπ;ρμTheremainderoftheprooftoget(6.3)remainsunchanged.Becauseofthezeroatthefrontof(6.4),wehaveignoredfactorslike1/2!or1/3!Noticealsothatonedoesnotneedtousenormalcoordinates.

220202CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESEinstein’sequationsread,uptoaconstantfactor,μνμνG=T,(6.5)whereGμνrepresentsthecomponentsoftheEinsteintensorandTμνistheenergy-momentumtensorcontributedbyallinteractionsexceptthegravita-tionalone.From(6.3)and(6.5),wegetμνT;ν=0.(6.6)Equation(6.6)statestheconservationlawofenergy-momentumofalltypesexceptgravitational.But,again,ithasverylittlesignificanceintheabsenceofconstantframefields.Gravitationalenergy-momentumremainsamysteryinEinstein’stheoryofgravityundertheLCC.10.7EINSTEIN’SDIFFERENTIAL3-FORMThesocalledenergy-momentumtensorisabadconcept.Itshouldbereplacedwiththeconceptofenergy-momentumvector-valueddifferential3−form,orsimplyenergy-momentumdifferentialform.Itisa3−form(inspacetime!)be-causeweintegrateitonvolumes.Itisvector-valuedbecausetheresultoftheintegration(whenitmakessense)isa(four)vector.Thecurrentofchargeisascalar-valueddifferential3−forminspacetime,misleadinglyrepresentedasa(four)vector(seesection7ofchapter4).TheHodgedualofthedifferential3−formcurrentofchargeinthealgebraofdif-ferentialformsisadifferential1−form.Wecanassociatewithitavectorfield,calledthecurrent(four)vector.Theenergy-momentumtensoristotheenergy-momentum(four)vectorwhattheelectromagneticcurrentistothechargescalar.Thosecurrentsgeneralizethedensitiesof(four)momentumandcharge.IntheEinsteintensor,thefirstindexreferstothecomponentsoftheenergy-momentum4-vector.Theotherindexconstitutesadisguisedformofstatingtheindicesofdifferential3−formsin4-Dbystatingwhichindexismissing.Equation(6.5)reallysaysthatavector-valueddifferential3−formconstructedfromRiemann’scurvatureisequaltoenergy-momentum3−form.Cartan[8]presentsuswiththecomponentsof“Einstein’svector-valued3−form”GasG≡Gμe,(7.1)μwhereG0≡ω1∧Ω23+ω2∧Ω31+ω3∧Ω12,(7.2a)G1≡ω0∧Ω23+ω2∧Ω30+ω3∧Ω02,(7.2b)G2≡ω0∧Ω31+ω3∧Ω10+ω1∧Ω03,(7.2c)G3≡ω0∧Ω12+ω1∧Ω20+ω2∧Ω01.(7.2d)

22110.7.EINSTEIN’SDIFFERENTIAL3-FORM203Cartan’sargumenttogettothoseexpressionsisratherabstruse,involvedasitiswithinvariantsofaquadraticsymmetricdifferentialform.Noticethattheindicesonthelefthandsidesaretheindicesmissingontherighthandside.Noticealsothethreecyclicpermutationsontherighthandsideofeachoftheseequations,aswellasthecyclicpermutationsoftheindices(1,2,3)ingoingfromG1toG2andthentoG3.WeshallnowseeGfromaperspectiveofdifferentialformsthattakevaluesinthetangentCliffordalgebraofspacetime.Itrequiressomeextraeffort.Readersfamiliarwithgammamatricesshouldhavenodifficultytofollow.μμRecallthatthesubscriptνinΩνandRνλπcorrespondstoalinearfunctionofvectors,whichthemetricallowsustoreplacewithavectorfield.WethusviewRiemann’scurvatureasνλ1νλ1νλνλ=Ω(eν∧eλ)=Ωeν∧eλ=Ωνλe∧e=Ωνλ(e∧e),(7.3)22wheretheexteriorproductmatchestheskew-symmetryofthetwodisplayedin-νλdicesinΩ.EinsteinessentiallybuiltwiththeRiemanniancurvatureavector-valueddifferential3−form.Forthat,onehastogetthe3−formsinEqs.(7.2)νλfromthe2−formsΩ.Inotherwords,onehastogetthevector-valuedGfromthebivector-valuedcurvature.Considerthevector-valueddifferential3−formdP(∧,.),wherethesymbol∧in(∧,·)isfortheexteriorproductinthealgebraofdifferentialforms,andthedotproductisforthetangentalgebra.ReaderscanverifywithcomputationssimilartothoseabouttofollowthatdP(∧,.)doesnotyieldtheEinsteindifferential3−form.However,letZdenotetheunite0eiejek(=e0∧ei∧ej∧ek)ofgradefourinthetangentCliffordalgebraofspacetime.Zisto(thetangent)Cliffordalgebrabasedon(e0,e1,e2,e3)whatwistoaCliffordalgebrabasedon(dx,dy,dz)(Seesection4ofchapter6).ThusthedifferentialformZistheHodgedualofinthistangentalgebra.Wenowhave(e0)2=1,(ei)2=−1.WeshallshowthatG=dP(∧,·)(Z).(7.4)Butwhyshouldonecareaboutanexpressionlikethis?Weshallseeattheendofsection7howeasilytheconservationlawofGisaconsequenceof(7.4).Anotherreasontocareisasfollows.ThedifferentialformforanEinsteintensorinndimensionswouldbeavector-valued(n−1)-differentialform.Butwestillwantenergy-momentumtobeavector-valueddifferential3−formfor3-volumeintegrationinhigherdimension(WehaveinmindaKaluza-Kleinspacewherepropertimeplaystheroleoffifthdimension).TheexpressiondP(∧,·)(Z)yieldsavector-valueddifferential3−form,independentlyofthevalueofn.Finally,inordertodealwith(7.4),onehastouseCliffordalgebra,thustakingusintherightdirectionifonewantstounifygravitationwithquantumphysics.Wereturntothemainargument.From(7.3),wereadilygetZ=Ω0ieZ+ΩijeZ=−Ω0iejk+Ωijek0,(7.5)0iij

222204CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESwhereeijisdefinedaseiejandthusasei∧ejinorthogonalbasisfields.In(7.5),(i,j,k)isanycyclicpermutationof(1,2,3).Thereissummationoverthethreepairsoftheform(ij)andthethreetriples(ij,k).From(7.1),(7.4)and(7.5)followsthatGeμ=ωμe(∧,·)(−Ω0iejk+Ωijek0).(7.6)μμG0isthengivenbyG0=G=ωle(∧,·)Ωijek0=ω1∧Ω23+ω2∧Ω31+ω3∧Ω12,(7.7)0lwhichis(7.2a).Theproductse·e0i,e·ejiande·ekiyieldei.Hence,readingfrom(7.6),0jkweformthecombinationω0e(∧,·)Ωjkei0+ωje(∧,·)Ω0keij+ωke(∧,·)Ω0jeik,(7.8)0jkwithoutsummationoverrepeatedindicesorcyclicpermutations.Byusingfor(i,j,k)thethreecyclicpermutationsof(1,2,3),weobtainthenegativesof(7.2b)-(7.2d)multipliedbye1,e2,e3respectively.Thesethreeunitvectorschangesignwhenloweringindices.Equation(7.4)havethusbeenshowntoyield(7.1)-(7.2).Atthispointandinordertoincreasefamiliaritywiththealgebrainvolvedhere,weprovidesomesimpleconceptsofthesame.Theidentity11dxdy=(dxdy+dydx)+(dxdy−dydx)(7.9)22appliestoanytypeofproduct,forinstance,tothetensorproduct(symbolnotexhibitedduepreciselytothegeneralvalidityof(7.9)).TheCliffordproductistheonewhere1(dxdy+dydx)isidentifiedwiththedotproductand1(dxdy−22dydx)withtheexteriorproduct.dxanddyaresaidtobeofgradeone,i.e.differential1−forms.Theirdotproductlowersthegradeandtheirexteriorproductraisesthegrade.Ongeneraldifferentiablemanifolds,wedonothaveCartesiancoordinates.We,however,haveωμων=ωμ·ων+ωμ∧ων,(7.10)withμν1μννμμν1μννμω·ω≡(ωω+ωω);ω∧ω=(ωω−ωω).(7.11)22Iftheωμarethosethatorthonormalizethemetric,theysatisfyρρω·ωμ=δμ.(7.12)WeshallseeinthenextsectionthatthecomponentsoftheEinsteintensorarethecomponentsGαμoftheHodgedualGzofGintheCliffordalgebraofdifferentialforms(calledK¨ahleralgebra)ofspacetime,i.e.Gz=Gαμeω=Gαeωμ,(7.13)αμμα

22310.8.EINSTEIN’S3−FORM:PROPERTIESANDEQUATIONS205where00123z=ωw=ωωωω,(7.14)andwhere,again,wehaveadaptedtheexpressionforwtotakeintoaccountthattherearenotCartesiancoordinatesystemsongeneralmanifolds.HencezistoaCliffordalgebrabasedon(ω0,ω1,ω2,ω3)whatZistoaCliffordalgebrabasedon(e0,e1,e2,e3).Regardlessofwhetherthesignatureis(1,-1,-1,-1)or(-1,1,1,1),z2is−1.Atthispoint,werelatetheGαμtothecomponentsoftheEinsteindifferential3−form,componentsdefinedimplicitlybyG≡Gα(ωμ∧ων∧ωπ)e≡Gα(ωμνπ)e.(7.15)μνπαμνπαFrom(7.13)and(7.15),wegetGαμω=Gα(ωμνπ)z.(7.16)μμνπTheproductofωμνπwithzpickstheelementofthebasisofdifferential1−formswiththeonlyindexρnotpresentinμνπ.Weputμνπinsuchanorderthat(μνπρ)beanevenpermutationof(0,1,2,3).From(7.12)and(7.16),wethenhaveαρραμαρμνπαμνπρG=ω·Gωμ=Gμνπω[(ω)z]=−Gμνπωωz.(7.17)Wenextusethatωμνπωρis±zandthatz2is−1.TheidentificationGαρ=Gα(7.18)μνπresults.10.8Einstein’sdifferentialform:properties,componentsandEinstein’sequationsTheEinsteintensorisdefinedasGαμe⊗ewiththesamecomponentsastheαμvector-valueddifferential1−formGz.Inthissection,weshalldealwiththefollowingsubjects.First,weshallshowthesymmetryGαρ=Gρα(8.1)withoutresorttothespecificformofGαμintermsofthecomponentsofRie-mann’scurvature.Thisreflectsthefactthat,ingeneral,thereshouldnotbeanyneedtocomputecomponentstoderivethesymmetry(8.1).Second,weshallcomputethosecomponentsbyexplicitcalculationsothatreaderswhocouldnotfollowalltheprecedingargumentsdonotharboranydoubtsthattheGαρderivedintheprevioussectionarethesameasinthetensorcalculus.ThatwillalsogiveuspracticewithCliffordalgebra.Wethendealwiththeconservationlawfromtheperspectiveofdifferentialformsandalsoshowthat,underthe

224206CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESLCC,theannulmentofthecovariantderivativeoftheEinsteintensorisequiv-alenttotheannulmentofdG.Finally,wewritethebynowtrivialexpressionforEinstein’sequationsintermsofdifferentialforms.Theproofof(8.1)appliestosimilarlyconstructedsecondranktensorsfromvector-valueddifferential3−forms.Readerstowhomthisalgebramayseemforbiddingshouldjustglanceattheproofwithoutstrenuousattemptatunder-standing.WewriteGαρasGαρ=Gβδαηρμ=ωρeα(·,·)Gβeωμ,(8.2)μβμβanduse(7.13)and(7.4)toobtainGαρ=ωρeα(·,·)(Gz)=ηωρeα(·,·){ωμeμ(∧,·)(Z)z},(8.3)μwhereη=(1,−1,−1,−1)or(−1,1,1,1).SinceZequals(1/2)ReνλωπσZ,μνλπσwefurtherhave1[ωμeμ(∧,·)Z]z=Reμ·(eνλZ)[(ωμ∧ωπσ)z].(8.4)νλπσ2AformulainCliffordalgebrastatesthatμπσμπσ(ω∧ω)z=ω·(ωz).(8.5)Itwouldbeoutofplacetogohereintotheproofofthisformula.Sufficeaninformalverification.Ifμequalseitherπorσ,bothsidesareevidentlyzero.Ifμisdifferentfrombothπandσ,ωμπσzisωρ,whereωρisthemissingindex.Ontheotherhandωπσzcontainsonlythetwoindicesotherthanπandσ,i.e.μandρ.Thedotproductwithωμyieldsωρ.Returningtothemainargument,wehave,from(8.3)and(8.4),1Gαρ=Reα·eμ·(eνλZ){ωρ·[ωμ·(ωπσz)]},(8.6)μνλπσ2and,usingthesymmetryRνλπσ=Rπσνλ,weget1Gαρ=Reα·eμ·(eνλZ){ωρ·[ωμ·(ωπσz)]}.(8.7)μπσνλ2Weexchangetheindicesαandρin(8.8)andinverttheorderofthetwoparen-thesestoobtain1Gρα=Rωα·ωμ·(ωνλZ){eρ·[eμ·(eπσz)]}.(8.8)μνλπσ2Sincethecontentsofallthesecurlybracketsarescalars(actuallythenumbersoneorminusone),therighthandsidesof(8.7)and(8.8)arethesameandthesymmetry(8.1)follows.

22510.8.EINSTEIN’S3−FORM:PROPERTIESANDEQUATIONS207WenowshowwhattheGαρwedefinedintheprevioussectionareintermsofthecomponentsofRiemann’scurvature.Multiplicationof(7.1)byzandcomparisonwith(7.13),yieldsGαωμ=Gαz,and,therefore,μGα=ω·(Gαz).(8.9)ρρConsider,forinstance,G0.WestartbyexpandingG0,givenby(7.2a):0G0=ω1∧[R23ω0∧ωi+R23(ωi∧ωj)]+ω2∧[R31ω0∧ωi+R31(ωi∧ωj)]+0iij0iij+ω3∧[R12ω0∧ωi+R12(ωi∧ωj)].(8.10)0iijIfρin(8.9)istobezero,G0zmustcontainω0asafactorfornon-nullmultiplica-tionwithω.ThusG0mustnotcontainω0.Thateliminatesfromconsideration0manytermsin(8.10).Ontheotherhand,thevaluesofiandjaredeterminedbythefactoratthefrontofeachsquarebracketin(8.10),sinceallthreeindicesmustbedifferentforatrivectornottobezero,andallthreeofthemarespa-tialindices.Wechoosetheorderof(i,j)sothatwegetcyclicpermutationsof(1,2,3).WethusgetG0=(R23+R31+R12)ω1∧ω2∧ω3.(8.11)233112Wecanrewritezasωωωω0andthenas−ωωωω0sothattheequation123321(ω1∧ω2∧ω3)z=−ω0becomesobvious.WethusgetG0=ω·[−(R23+R31+R12)ω0]=−(R23+R31+R12).(8.12)00233112233112ForG11,wetakeintoaccount(7.18)andthusconsiderthetermsω0∧ω2∧ω3ontherighthandsideof(7.2b),i.e.inω0∧[R23ω0∧ωi+R23(ωi∧ωj)]+ω2∧[R30ω0∧ωi+R30(ωi∧ωj)]+0iij0iij+ω3∧[R02ω0∧ωi+R02(ωi∧ωj)](8.13)0iijtogetG11=(R23+R30+R20)=−G1.(8.14)2330201WesimilarlygetG01=−(R02+R03)=-G0(8.15)12131andG12=−(R10+R13)=G1.(8.16)20232TheothercomponentsofEinstein’stensorcanbereadalmostdirectlybysimplyapplyingcyclicpermutationstothelasttwo.NoticethatwehavereachedtheEinsteintensorwithoutgoingthroughtheRiccitensorReν⊗eμ≡Rλeν⊗eμ.(8.17)νμνμλTheconservationlawoftheEinstein3−formtakesjustthefollowingonelinecomputationwhenoneusesK¨ahler’scalculus.Weindeed,havedG=d[dP(∧,·)(Z)]=dP(∧,·)[(d)Z]=0,(8.18)

226208CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESsinceZisaconstant0−form,and,fortheLCC,d(dP)iszero.Thatallowsustoignoretheeffectofdonanythingherebut.Wehavealsousedd=0.InordertoshowthattheannulmentsofGαρanddGareequivalent,we;ρchooseframefieldsthat,atanygivenpoint,annultheLCC.Inthatcase,Gαρ;ραρbecomesG.Ontheotherhand,weresortto(7.15)toobtaindG.Inthe/ρsameframefields,wehavedG=dGα(ωμνπ)e=dGαρ(ωμνπ)e=Gαρ(ωρμνπ)e,(8.19)μνπαα/ραsincedeisdirectlyzero,anddωμνπiszerobecauseeachdωβsatisfiesdωβ=αωγ∧ωβ=0.Hehavealsoresortedto(7.18).HencedG=0isequivalenttoγGαρ=0inconstantframefields(HadtheconnectionnotbeenLevi-Civita’s,;ρwecouldnothavesetdωνequaltozero,obstructedaswewouldhavebeenbythetorsionterminthefirstequationofstructure).Sincetheresultistensorial,itisvalidinanyframefield.Theproofisnowcomplete.Finally,inviewofalltheabove,wewriteEinstein’sequationswithdifferen-tialformsasG=Π,(8.20)whereΠisthevector-valueddifferential3−formwhichisimproperlytreatedasatensorandknownastheenergy-momentumtensor.Πtakesmanyspecificformsdependingonwhatpartsofthephysicsitcomesfrom.10.9TheEinsteinequationsforSchwarzschild’smetricwiththreearbitrarycoefficientsIntheliterature,theSchwarzschildmetricisfirstwrittenwiththreearbitraryfunctionsowingtosphericalsymmetry.Theyarethenreducedtotwobyvirtueoftheoptiontochoosetheradialcoordinate.Thisimpliesthatthelefthandsideof,say,thethirdEinsteinequationforthepointmass(afourthoneisarepetitionofthethirdonebecauseofthatsymmetry)canbebuiltintermsofthelefthandsidesofthefirsttwo.Sucharelationwouldhavetobematchedbythesamerelationontherighthandsidesinproblemswherethesewerenotzero.Wethussettosolvethepointofmassproblemwiththreearbitraryfunctions.Whybringthisup?Wewishtostartpreparingthewayforsolvingsophis-ticated,non-traditionalproblemsinvolvingEinstein’sequations.Suchwouldbethecase,forinstance,withamoresophisticatedversionoftheReissner-Nordstromproblem.ToquoteEinstein(p.93of[29]):“ButnoreasonablepersonbelievesthatMaxwell’sequationscanholdrigorously.Theyare,insuit-ablecases,firstapproximationsforweakfields.”Atthispoint,wedonothavesomuchinmindadifferentsetofMaxwell’sequations,butapotentiallydifferentelectromagneticenergy-momentum3−form.Einstein’scommentstillapplies,sinceachangeinfieldequationswouldlikelyentailachangeinelectromagneticenergy-momentum.

22710.9.EINSTEINEQUATIONSFORSCHWARZSCHILD209Wewritethemetricasds2=e2ν(r)dt2−e2μ(r)dr2−e−2λ(r)r2(dθ2+sin2θdφ2).(9.1)Theindependentnon-vanishingαν’spertainingtoabasisofωμ’sthatorthonor-μmalize(9.1)arereadilyobtainedbyinspection:α0=νeν−μdt,α2=e−μΛdθ,11(9.2)α3=eλ−μΛsinθdφ,α3=cosθdφ,12whereΛ≡λr+1,andwhereprimesareusedtodenotederivativeswithrespecttor.Thecomponentsofthecurvature2−formfollow:1ν−μΩ0=e[ν+ν(ν−μ)]dr∧dt,Ω2=−νeλ+ν−2μΛdt∧dθ,0Ω3=−νel+ν−2μΛsinθdt∧dφ,0Ω2=eλ−μ[(λ−μ)Λ+λr+λ]dr∧dθ,132λ−2μ2Ω2=sinθ[−1+eΛ]dθ∧dφ,Ω1=eλ−μ[(λ−μ)Λ+λr+λ]sinθdr∧dφ.3WeproceedtocomputetheEinstein3−form.Asaconsequenceofsphericalsymmetry,G3doesnotyieldanythingthatisnotalreadycontainedinG2.Itwillbeomitted.Thenotationdrθφmeansdr∧dθ∧dφ,andsoon.Wethusobtain:ω1∧Ω23+ω2∧Ω31+ω3∧Ω12=drθφeμ(1−e2λ−2μ{Λ2+2r[(λ−μ)Λ+Λ]}),(9.3a)023230302tθφν2λ−2μ2ω∧Ω+ω∧Ω+ω∧Ω=dsinθe(1−e(Λ+2rνΛ)],(9.3b)andω0∧Ω31+ω3∧Ω10+ω1∧Ω03==−dtφrsinθeν−μ+λ[Λ(λ−μ+ν)+(ν+ν2−νμ)r+Λ].(9.3c)Weequatetherighthandsidesoftheseequationstozeroandsimplifytoobtain:1−e2λ−2μ{Λ2+2r[(λ−μ)Λ+λr+λ]}=0,(9.4a)1−e2λ−2μ[Λ2+2rνΛ]=0,(9.4b)andΛ(λ−μ+v)+(ν+ν2−νμ)r+Λ=0.(9.4c)Fromthefirsttwooftheseequations,weget:Λ−1Λ=μ+ν−λ,(9.5)

228210CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESwhich,afterintegrating,yields−(λ−μ−ν)=lnΛ+constant.(9.6)HenceΛ=ec+μ+ν−λ,(9.7)wherecistheintegrationconstant.Wemultiply(9.4b)bye−2c−2νinordertoconvertthecoefficientofthebracketintoΛ−1.ThisallowsustowriteΛas2rνe2ν+2cΛ=.(9.8)1−e2ν+2cFrom(9.7)andthedefinitionofΛ,weget12νe2ν+2cλ+=,(9.9)r1−e2ν+2cwhichintegratedyields−λ2ν−2cre=(1−e),(9.10)r0whererisanotherintegrationconstant.Solvingthisforeν,weget0ν−λr01/2−ce=(1−e)e(9.11)rand,from(9.7),μλ−ν−ce=Λe.(9.12)Thelasttwoequationsyieldeμandeνasafunctionofλ,andbecomer0!1/2eν=1−,eμ=e−ν,(9.13)rforλ=C=0.Thisisaclassicresultingeneralrelativity.Itisworthnoticingthatequations(9.12)havebeenobtainedfrom(9.4a)-(9.4b)withoutresortto(9.4c),whichdoesnotcontributeanythingnewaswenowshowbydirectproofof(9.4c)from(9.4a)and(9.4b).Wedifferentiate(9.4b)andmultiplyby−1Λ−1e2(μ−λ)(Λ=0isnon-physical).2Using(9.5),whichwasimpliedby(9.4a)-(9.4b),wereplaceΛ−1Λintheverylasttermoftheequationsoobtained.Wethusget(λ−μ)Λ+(λ−μ)2rν+Λ+ν+rν+rν(μ+ν−λ)=0.(9.14)Weshallnowshowthat(9.14)coincideswith(9.4c).Equation(9.14)hassixfirstleveltermsonthelefthandside.Thethirdtermisthelastoneonthelefthandsideoftheequation(9.4c).Thesumofthefourth,fifthandsixthtermsof(9.14)canberewrittenasr(ν+ν2−μν)+ν−rνλ+2rνμ.(9.15)

22910.9.EINSTEINEQUATIONSFORSCHWARZSCHILD211Thetermwiththeparenthesisin(9.15)isanotheroneofthosepresentin(9.4c).Ifweaddtheotherthreetermsin(9.15)withthesecondtermin(9.14),wegetν(1+rλ).ThisequalsνΛ,whichaddedtothefirsttermin(9.14)yieldsthefirsttermin(9.4c).Henceequation(9.4c)isthedifferentiated(9.4b)multipliedby−1Λ−1e2(μ−λ).Thisimpliesacorrespondingrelationbetweentherighthand2sidesofEinstein’sequations,astrongconstraint.InEinstein’stheoryofgravitation,energy-momentumtensorsarebroughttothefieldequationsfromotherpartsofthephysics(Incidentally,notallenergy-momentumtensorsaresymmetric;Einstein’stensoris).Inahypotheticaluni-ficationoftheinteractions,theEinsteintensormightbejustatermatparwiththeotheronesinthesameequation,theseparationintoleftandrighthandsidesbeingartificial.Sphericalsolutionsatveryshortdistancemightnotevenexist.Theubiquitousspindoes,afterall,breakthesphericalsymmetry.Thefinalchapteronthetheoryofgravitation—whichcertainlyisintherighttrackwithEinstein’stheory—mayhavenotyetbeenwritten.

230212RIEMANNIANSPACESRiemannianpseudo-spacesUpdatedversionbyLevi-Civita(oldRiemanniangeometry)(Attachrighttolefthandside)CHAPTER10.RIEMANNIANSPACESANDPSEUDO-SPACESωμ∗de=e(x+dx)−e(x)μμμdωμ=ων∧αμμναν−→d¯P≡ωeμαμν+ανμ=0μd¯e≡ανeμμνd˙v∗=(dvμ+vναμ)eνμ0=d˙d¯P=(dωμ−ων∧αμ)eνμΩμ∗=dαμ−αλ∧αμd˙d¯e=Ωνvμeνννλμμνd˙d˙v=ΩνvμeμνdΩν+Ωλ∧αν−αλ∧Ων=0μμλμλ∗ωμdefinedbyds2=,=±1,Ων=Rν(ωλ∧ωπ).μμμμμλπd¯isnotanoperatorbutfirsthalfofsymbols¯dPand¯de.d:formaldifferentiation;d˙d˙=0.

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233Chapter11EXTENSIONSOFCARTAN11.1INTRODUCTIONDifferentialformsconstitutethemostpowerfullanguagefordifferentialgeom-etry.E.Cartanusedthemtodevelophis´theoryofconnectionsintheearly1920’s[11],[12],[14].Butthoseobjectshavealifeoftheirown—evenlargerthandifferentialgeometryitself—sincetheycanbeendowedwithstructureadditionaltowhatisrequiredinthatgeometry.Thepurposeofthisandthenexttwochaptersistoshowthatsomelargelyoverlookedmathematicsofthe19thand20thcenturieshelponeunderstandhowtofurtherextendthefieldofdifferentialgeometry.RiemannandFelixKleinarethetwofatherfiguresintheevolutionofge-ometryinthe19thcentury(seeSection8.1).Butthebigpictureisincompletewithoutathirdfatherfigure,namelyHermannGrassmann([43]isapublicationinEnglishofhismainwork;seeappendixBforabiographicalsketch).Hisideasweresoadvancedforhistimethattheirrelationshiptogeometrystillislargelymissed.HisrelevanceissuchthatwehavefounditpertinentintroducinghisbiographyinAppendixB.TherelevancefordifferentialgeometryofGrassmannwork,whichispre-dominantlyalgebraic,liesinthatthereisonlyonepointofseparationbetweenalgebraandgeometry.Althoughazeroisabsentinaffinespace(studiedbygeometry),thelattercanbeidentifiedwithavectorspace(studiedbyalgebra)ifwechooseanarbitrarypointtoplaytheroleofzero.Buteventhe“zeropoints”arenotarbitraryindifferentialgeometry,sincetheyarethepointsoftangencyoftangentspacestodifferentiablemanifolds.Onecannot,therefore,givetoomuchimportancetotheseparationofalgebraandgeometry.Inascenariowheredifferentialgeometrywereamainfieldofresearchbutonlyrudimentsofalgebrawereknown,thelatterdisciplinewouldhaveemergedasabranchofdifferentialgeometry.Sincetherearebranchesofalgebrafar215

234216CHAPTER11.EXTENSIONSOFCARTANremovedfromdifferentialgeometry,wemayviewthelatterasthesectorofthealgebra-geometryedificethatconcernsitselfwithdifferentiationsandintegra-tions,i.e.theoperationsthatallowonetogeneralizetheelementaryorKleingeometries.Thelogicalandhistoricalorder,however,isthatextensionsofal-gebraleadtoextensionsoftheconceptofdifferentiation,whichinturnleadtoextensionsofstandarddifferentialgeometry.OnesuchextensionleadstoFinslerbundlesand,relatedtothem,toacanon-icalKaluza-Klein(KK)theorythatconvertsthetheoryofmovingframesintosomethingricher.InFinslerbundles,thepairofgroups(G,G0)isreplacedbyatripleofthem(G,G0,G00).Thus,forinstance,thespecialrelativisticspace-timestructurewouldbeseenasrepresentedbythePoincar´e,LorentzandSO(3)groups.InKKtheory,pointsplayaroleadditionaltotheonetheyplayinthetheoryofmovingframes,whichusethepointsofthemanifoldtoanchorthemselvestoit.Thisisapassiverole.IntheKaluza-Kleinextensionofwhichweshallspeakinthischapter,pointsalsorepresentpointparticlesthatmovearoundindepen-dentlyofanyframes.Inthisway,onedoesnotneedtoviewthetrajectoryofaparticleasasuccessionoforiginsofframes,asisthecaseinthetheoryofthemovingframe.So,Gplaysheretheroleofsubgroup.AnotheralgebraicextensionwithdifferentialgeometricimplicationswouldbethereplacementofaderivativebasedonexterioralgebrawithonebasedonCliffordalgebra.Inparalleltoour“introducing”theconceptsofaffinecurva-ture,torsionetc.throughexteriordifferentiationofvectorfields(Seesection5.7),wecouldconceiveinprincipleamorecomprehensivegeometrybasedonthereplacementofexteriordifferentiationwithaClifford-algebra-baseddiffer-entiation.Also,linearalgebraandLiealgebrawouldbeimmersedinCliffordalgebra.ThiswouldbeastillunexploredK¨ahlergeneralizationofthetheoryofmovingframes.Finally,thereisanotheraspecttotheKKextensionofgeometry:oneshouldviewsymmetriesinanewlight.The5-Dstructurenowisthemainone.Thegroupsofsymmetrycannotbethesameinsubspacesthatappearsimilarbutarenotactuallysoinaprofoundway.Oneshouldlookatspacetimesymmetryasamanifestationinspacetimesubspaceofsomesymmetrypertainingtothe5-DKKspace.Themanifestationofthatsymmetrymaytakeanotherforminspace-propertimesubspace.Wedealwiththatinthelastsectionofthischapter.11.2Cartan-Finsler-CLIFTONInthisbook,wehaveseentheaffineandEuclideanbrandsofdifferentialgeom-etryastheoryofmovingframes.TheirKleingeometriesareostensiblyrelatedtocorrespondingfiniteLiegroupsandalgebras.Theframebundlesareofdi-mensionn+n2andn+[n(n−1)/2],respectivelyintheaffineandEuclideancases.Incontrast,thecrucialdimensioninthetensorapproachtodifferentialgeometryisnanditsintegerpowers,bothintheaffineandEuclideancases.Thisisrelatedtothefactthatthetensorcalculusimplicitlydealswithwhat

23511.2.CARTAN-FINSLER-CLIFTON217weknowtobesectionsofframebundles,butwherethelastonesareotherwiseabsent.ThetensorcalculusprecedesCartan’stheoryofconnectionsbytwolongdecades.Butworseisthefactthattheabsenceofframebundleconsiderationsalsooccursinmostmodernapproachestotheclassicalsectorofdifferentialgeometry.Thus,thekeyelementinFinslergeometryisseentobe,incorrectlyinouropinion,Finslerianmetrics,notFinslerframebundles.Butgeometryiswhatgeometersdo.Irespectthatview,butitmaybeoflittleinterestinareasofgeometrythatshouldconcernphysicists.Ishalltrytoshowherethatthereisabetterway.Imustfirstreport,however,thelargeamountofpublicationsonsophisticatedphysicalissuesbyH.BrandtusingatypeofFinslergeometrythatdoesnotuseframebundles.Butitatleastrelatesthetangentbundletosophisticatedphysics.Thetitle“FinslerianQuantumFieldTheory”ofoneofhispapers[2]speaksbyitselfandcontainsthealmostcompletelistofhisFinslerrelatedpapers.Cartan’sseminalpaperontheCartan-Finslerconnection[19]usesthemov-ingframemethod,buttherelationtoaframebundleandtotheCartan-Kleinprogramisnotclear.AmoreexplicitapproachtothetypeofFinslergeometryweadvocateisanimportantpaperbyChernof1948onthissubject[25],butmuchlesssoinhisworkofthelasttwodecadesofhislife.Thismayhavebeenduetotheevolutionofhisinteresttowardsglobalissues,whichareconnection-independent.Thepresentauthor’sapproachtoFinslergeometryfallswithintheCartan-Kleinprogram.Itsubordinatesthemetrictotheconnection,andisagainbasedonthemovingframemethod[73],[74],[75],[76].SeealsoDeicke[30],[31].TheconceptofFinslerconnectioncanbeintroducedinaffinegeom-etry[73],i.e.independentlyoftheconceptofmetric.Asaby-product,wemayhaveFinslerconnectionsonRiemannianandpseudo-Riemannianmetrics.Thebreakthroughonaffine-FinslerconnectionsisduetoClifton,withprod-dingbythepresentauthor.Ibroughttohisattention,atatimewhenhehadacquiredadeepunderstandingoftheCartan-Finslerconnection[19],thathisFinslerpracticeoverlookedthephilosophyoftheCartan-Kleinprogram.InCartan’sapproach,thecharacterofageometryisgrantedtoitbytheconnec-tion(remembergeometricequality),notbythemetric(rememberRiemannianpseudo-spaces).AndthereisnotamajorreasoninphysicstoconsideratthispointmetricsotherthantheLorentzmetric,eveniftherewereaphysi-callysuitablepreferredframeandtheconnectionofspacetimewereFinslerian.IchallengedCliftontofirstintroduceFinslerianaffineframebundles,whichwouldthenberestrictedtoobtainbundlesforFinsler-Euclideanconnections.Hedelivered[73](readerscanfindinthatreferencewhyCliftonshouldhavebeenthemainauthorof[73]-[75]butwasnot).Themathematicalsignificance(oratleasttheutility)ofFinslerframebun-dlegeometryliesinthatitbringsupthequestionofwhatistheelementarygeometryunderlyingtheCartan-CliftonbranchofFinslergeometry.InspectionoftheformaldefinitionofCliftonofaffine-Finslerconnections[73]leadsonetoconcludethat,asalreadysaid,thereisatripleofgroups,andnotjust(G,G0).Thephysicalsignificanceoftherestrictionoftheaffine-Finslerbundleto

236218CHAPTER11.EXTENSIONSOFCARTANpseudo-RiemannianmetricsofLorentziansignatureisthatthegeometryofthespacetimeofspecialrelativityemergesastheKleingeometryunderlyingconnec-tionsonthosebundles.Saidbetter,thereisawaytolookatsuchaspacetimeotherthanasbeingpseudo-Riemannian.Thisinturnyieldsanewwayoflookingat4-velocities,whichiscrucialfordevelopingKKtheorycanonicallygeneratedbyLorentz-Finslergeometry.ThisinturnbringsupthesubjectofcanonicalKKgeometryforhighenergyphysics(Seenextsection).Yang-Millstheoryisbasedonauxiliarybundles,notdirectlyrelatedtothetangentbundle.Unfortunately,modernapproachestotheclassicalsectorofdifferentialgeometryforphysicistscarelittleaboutframebundles.ThisauthorcaredandfoundthattheLorentzforce(abstractionmadeofproportionalityconstants)isunavoidableintheFinslerframebundle.OnedoesnotevenhavetolookfortheconnectionwhoseautoparallelsaregivenbytheequationofmotionwithLorentzforce;itisunavoidableintheFinslerframebundlewithmetricofLorentzsignature.Inadditiontothegravitationalforcecontributedbythemetric,Lorentz-Finslerconnectionscontributewithanacceleration(thusforce)thattakestheformof—andonlyof—theLorentzforcewhenonlytheleastexplicitlyFinsleriantermsintheequationsofstructureareconsidered[79],[81].11.3Cartan-KALUZA-KLEINConnectionsonFinslerframebundlesrepresentgeometryasatheoryofmovingframeswithvelocitieslocatedonthebasespaceratherthanonthefibersofthestandardframebundles.ThisfeaturehelpstosolveaproblemwhichmaybesaidtohavebeenintimatedbyCartan,thoughhemayhaveoverlookedreturningtoit.Weproceedtoexplain.InhispaperonEinstein’sequations[8],Cartanobtainedtheequationsofstructureof3-DEuclideanspaceasconsequenceofthefollowing.Hehadaparticleandaframetowhichhereferredthecoordinatesoftheparticle.Heleftthepointfixedandcombinedadifferentialtranslationandadifferentialrotationoftheframe.Hedifferentiatedtheequationthatstatesthedifferentialsofthecoordinatesofafixedpointintermsofthattranslationandrotation.Furthermanipulationallowedhimtoobtaintheequationsofstructure.Movingthepointwasnotconsidered.Ifonedid,anequivalenceofactiveandpassivetranslationsmightbeinvokedinEuclideanspace,butnotingeneral.Beasitmay,themotionoftheparticleinconnectionwiththatderivationbringstotheforethattheparticlemighthavearoletoplayintheequationsofstructure,whichisnotpresentlythecase.ThemotionofparticlescanbeintegratedintothecoreofgeometrythroughaKKspacewherethepropertimeoftheparticleconstitutesthefifthdimension.Theformer4-velocityisthentheunitvectorassociatedwithit.Notsurpris-ingly,Cartanhimselfintroducedin1923aformulationofseveralbranchesofthephysicsintermsofafivedimensionalspace,thedimensionalitybeingdisguisedbyhisuseofPl¨uckeriancoordinates[11].Hisbivectorscomeintwoclasses.Six

23711.3.CARTAN-KALUZA-KLEIN219ofthemhecallsbivectorsandbelongtothespacetimesubspace.Theotherfourhecallsslidingvectors,whichintimatestheassociationofthefifthdimensionwiththeparticle.See[77].This“canonical”KKspacehasverysignificantfeatures,whichseemstoindicatethatthereisstillmuchtolearnabouttheflatspacetimeofspecialrelativity.Lete0denotetheunitvectorinthelaboratory’stimedirection,andletudenotetheunitvectorinthepropertimedirection.Considerthesubspacesspannedby(e0,e1,e2,e3)and(e1,e2,e3,u).Inspecialrelativity,thespacetimebasesareorthonormalbutthebases(e1,e2,e3,u)ofspace-propertimesubspacesarenotorthogonalsincethedotproductsei·umustbeuiγ,whereγis(1−u2)−1/2.Needlesstosaythat,inthisKKperspective,thedynamicsofaquantumsystemwillconcernthesubspacespannedby(e1,e2,e3,u),ratherthantheonespannedby(e0,e1,e2,e3).ThelackoforthonormalitydoesnotbodewellforspecialrelativityinthisKKcontext(Onecanalwaysorthonormalize,butthereplacementforuwillnolongerresultinatangenttothetrajectoriesofparticles).Butthereisasurprise.ConsiderthepreferredframealternativetospecialrelativitywithabsoluterelationofsimultaneitythatstillcomplieswiththekeyexperimentalresultsofMichelson-Morley,Kennedy-ThorndikeandIves-Stilwellexperiments[84].Thetimelikeunitvectorisnolongerperpendiculartothe(e,e,e)subspaceof123thisalternative,butthevectoruis,likee0isorthogonalto(e1,e2,e3)intherelativisticcase.InthisKKscenario,comparisonofthespace-propertimesubspaceforSRandforthejustmentionedalternativeappearstoindicatethatthegoodworkingoftherelativisticquantummechanicsthatisbasedontheorthonormalbases(representedintherelationsamongthegammamatrices)appearstobeanargumentinfavorofpreferredframephysicsratherthanspecialrelativity.Sincespecialrelativityworkssowell,anypreferredframeproposalwillap-pearunattractiveinprinciplegiventheconcomitantnon-orthogonalityofeto0(e,e,e)inthepreferredframescenario.Thatthisisnotaproblemhasbeen123wellunderstoodbythosewhosupportthethesisofconventionalityofsynchro-nizations,whichstatesthatspecialrelativityandthepreferredframealternativebeingconsideredareindistinguishablefromeachother.Thoughconventional-istsmayhavecarriedtheirclaimtoofar,thereisneverthelessmuchtruthinit.Inthepreferredframescenario,onecannotholdasynchronizationconsistentwiththespacetimetransformationswithabsolutesimultaneity[84].Clocksinourrealworldarenotatrestorinslowmotionwithrespecttoahypotheticalpreferredframe.TheyareautomaticallysynchronizedasperEinstein’ssyn-chronizationprocedure.Thefailuretoholdanabsolutesynchronizationdoesnotinvalidatethepossibilityofaphysicsbasedonanabsoluterelationofsi-multaneity,butmasksmostorallofitseffects.Formoredetailsandreferencessee[84].ThereareseveralotherpowerfulreasonsadvocatingtheaforementionedKKspace.Letuspointhereataphysicalone.ThelastsentenceoftheprevioussectionspeaksofthepossibleassociationofelectromagneticfieldwithtorsionthroughautoparallelsinFinslerframebundles.Letvμbethecomponentsof

238220CHAPTER11.EXTENSIONSOFCARTANthe4-velocityofaparticle.Asstatedattheendofsection9.7,atorsionoftheformΩμ=−vμFisanincorrectwayofwritingwhatshouldbeconsideredasanexpressionpertainingtotheFinslerframebundle,wherevμisnotafieldonacurvebutredundantcoordinatesontheFinslerianbasemanifold.Thevμbecomethecomponentsofthe4-velocityonnaturalliftingsofspacetimecurves.Morespecifically,the4-velocityise0intheFinslerbundle,aninvariantundertheSO(3)groupofthefibersofthatbundle.WetranslatethisfindingtotheKKspace.FbecomesthecomponentofthetorsioninthedirectionofuintheKKspace.But,asinstandardKKtheory,uisnotjustonemorevector;thefifthdirectionisdifferent.KKspaceisunlikespacetime,wherealltransformations(translationsandisometries)notinvolvingnulldirectionsareonanequalfootinguptoissuesofsignature.Transforma-tionsinaKKspaceinvolvethefifthdirectiondifferentlyfromtheothers.InourKKtheory,boostsofparticlesarenolongertheisometriesknownashyper-bolicrotationsinspacetimebutintime-propertimesubspaceoftheKKspace(thereisnoproblemwiththethreeindependentcomponentsofthevelocity;theconnectiontakescareofthat,asitdoesinFinslergeometrywhereduisde0).Cartan’stheoryofmovingframesisonceagainamagnificenttooltoexploreextensionsofclassicaldifferentialgeometryofinterestforphysics.11.4Cartan-Clifford-KAHLER¨Thisbookmusthavemadeclearbynowthatdifferentialgeometryisaboutexteriordifferentiation.Moreprecisely,itiscalculusofvector-valueddifferen-tialformswheretheunderlyingalgebraofthedifferentialformsproper—i.e.disregardingtheaccompanyingvaluednessalgebra—isexterioralgebra.Ontheotherhand,theK¨ahlercalculusisbasedonCliffordalgebraofdifferentialforms.ExteriordifferentiationiscontainedinK¨ahlerdifferentiation,butnotcon-versely;thereislessinformationintheexteriorderivativethanintheK¨ahlerderivative.Andyetintegrabilityofasystemisbasedonlyonanintegrabil-ityconditionaboutexteriordifferentiationeventhoughthecomplementaryin-formationprovidedbyinteriordifferentiationisabsent.Itlooksasif,intheintegrationprocess,wegetforfreetheinformation(containedintheinteriorderivative)lostbyexteriordifferentiating.Anincipientstudyofthisissuewasundertakenelsewhere[80].Thatpaperintimatedthattwoalgebras,whatevertheirnature,aredeeplyintertwined,aswehavealsoseeninthisbook.Atthattime,wedidnotprovidespecificsofhowthattakesplace.Thedetailsarebecomingincreasinglyclear.Theexteriorproductoftwoconsecutivesidesofaparallelogramgivessomeinformationaboutthelatter.But,onacurvedsurface,parallelogramsintheordinarysenseoftheworddonotexist.Wemayintegrateascalar-valueddifferential2−formf(x1,dx2)dx1∧dx2onaregionofacurvedsurface,butthisinformationisnotequivalenttotheinformationcontainedintheexteriorproductoftwovectors.Thelatterspeaksinparticularofthespatialorientation

23911.5.CARTAN-KAHLER-EINSTEIN-YANG-MILLS¨221inEuclideanspaceofthefigureformedbythetwovectors.Inthecaseofthecurvedsurface,theaforementionedintegrationgivesonlyanumber,butnotanorientationinspace.Forthat,onewouldneedtointegrateabivectorvalueddifferential2−form.Considerasystemofdifferentialformequationsforaquantummechanicalstructurethathassymmetries.IfthedifferentialformsareendowedwithaCliffordratherthanaexterioralgebrastructure,onemayexpandmembersofthealgebraofdifferentialformsintermsofmembersoftheidealsthatthosesymmetriesdefineinthealgebra.Theimplicationsofsuchexpansions[48],[83]mayberelevantforthegrowthofdifferentialgeometry,aswenowexplain.Forangularmomentum,K¨ahlerbuildsaleftidealwiththeidempotent1(1+2idx∧dy)(beawarethat1(1+dx∧dy)isnotanidempotent).Butwedonot2needtheunitimaginaryifwetaketheidempotenttobe1(1+dx∧dya∧a),212alsowrittenas1(1+dxdyaa).Theconcomitantphaseshiftsthatgivethe212dependenceontheangularcoordinatearenowtobeviewedasexp[(1/2)φa1a2],where,again,a1a2playstheroleoftheunitimaginary.Similarly,theunitimaginaryintheenergyoperatorandintheexponentofphaseshiftsassociatedwithtimetranslationswouldbereplacedbytheunitvectorinthepropertime(time)directioninthespace-propertime(respectivelyspacetime)subspacesoftheaforementionedKKspace.EnterCartan.HewasnotquiteasexplicitwithregardstoCliffordalgebras,inpartduetothefactthattheappearsnottohavediscussedthemexceptinhislongpaperof1908onComplexNumbers[5](Incidentally,thisisaverygoodpapertolearnofthegreatvarietyofdifferentproductsintroducedbyGrassmann).Buthemadeanextensiveuseofexteriorvalueddifferentialforms,whichinvolvesalsoasecondexterioralgebra.IfhehadextendedthetwoexterioralgebrasinthoseformstoCliffordalgebras,hewouldhaveexpandeddifferentialgeometrybeyondtheexteriorcalculus,bringingidempotentsandidealsintotherealmofdifferentialgeometry.Arapprochementoftherespectivemathematicsofgeneralrelativityandofquantummechanicswouldhaveensued.Tosaytheleast,wehaveindicatedinthissectionhowonecanexpandCartan’stheoryofmovingframesintonewrealmsalongthelinessuggestedbytheworkofK¨ahlerthroughhisexploitationoftheCliffordalgebraofdifferentialforms.11.5Cartan-K¨ahler-Einstein-YANG-MILLSThetitlesoftheprevioussectionsinthischapterhadonlynamesofmathemati-cians.WeshallnowinvolvethenamesofEinsteinandYang-Millsforfurtherdevelopmentofdifferentialgeometrydirectlyrelatedtothetangentbundle.Coulditnotbethattheauxiliarybundlesofpresentdayphysics—thoughnotanyauxiliarybundlethatonecouldthinkof—aredirectlyrelatedtothetangentbundle?Andwhatwouldthatdifferentialgeometrybelike?WehaveintimatedinthisbookthattheFinslerframebundleallowsforthematchingofapieceofthetorsionwiththeelectromagneticfield.Thetorsion

240222CHAPTER11.EXTENSIONSOFCARTANand,moreinterestingly,thecontorsionhavemanymoredegreesoffreedomthanneededtoaccommodatethatfield,butapparentlynotinawaythatwouldinvolvetheU(1)×SU(2)×SU(3)group.Inotherwords,itisnotatallclearwheretheweakandstronginteractionsare.Certainlytheyarenotinspacetimesymmetryiftreatedinthetritewayinwhichtheparadigmmakesuseofit.WeneedRiemann’scurvatureforgravitation.TheclueastohowtosatisfythisneedinawaycompatiblewithaconnectionotherthanLC’sliesinthestudyofconservationlawsthatweundertookinsection8.10.ItleadsonetotheconclusionthatthereisnottrueconservationongeneralizedmanifoldswithoutTP.UnlesswehavetrivialTP(likeinKleingeometries),thetorsionandthusthecontorsion,β,arenotnull.SinceβisthedifferencebetweenωandtheLC’sdifferentialformα,itisvaluedinthesameLiealgebraastheyare.βisgoingtoplaytheroleoftheYang-Millsconnection.Itcertainlyisnotaconnection,whichisbetterthanifitwereone,beingasitispartoftheconnection.Theotherpart,α,representsthegravitationalinteractionandbothpartsarejoinedinaverynaturalandstrongway.Whatistheconnectionthatcanaccommodateallthat?AndwhathasallthistodowithEinstein?Inthelate1920’s,heproposedTPfortheconnectionofspacetime.Cartanadvisedhimthatheincorporateintohissystemofequa-tionsthefirstBianchiidentityandthecurvature.HadEinsteinhadabetterunderstandingofdifferentialgeometry,hewouldhaverealizedhowwelldidCar-tan’ssuggestionsfithisthesisoflogicalhomogeneityofphysicsanddifferentialgeometry[38].EinsteinabandonedhisTPprojectinfrustration.Cartandidnotgoonhisownand,inanycase,thestateofcalculusandofthegeometricartwasnotripeforthetask.Indeed,atthetimeofhiscorrespondencewithEinsteinonTP,CartanhadnotyetdonehisseminalworkonFinslergeometry[19],whichwascrucialforClifton’sacquiringtheframebundlevisionofwhichwespokeinthischapter.Inaddition,anotherelementwasneededintheopinionofthisauthor:theK¨ahlercalculusandconcomitantvisionofquantumphysics(Seechapter13).CartanneverthelessdiscussedthegeneralcharacteristicsofEinstein’sattemptatunificationwithteleparallelism[18].TheK¨ahlercalculusisneededfortheKKframework,whichinturnisneededforYang-Millsinthisgeometricperspective.KKspaceallowsoneforthesamebasicsymmetrytomanifestitselfasLorentzsymmetryinthespacetimesub-spaceandasU(1)×SU(2)symmetryininthespace-propertimesubspace.Inthissubspace,SU(2)wouldreplaceSO(3)becausewewouldbedealingwiththeactionofrotationsonspinorsasmembersofidealsofspace-propertimeCliffordalgebra.RegardingU(1),wealreadysawitsemergenceintheFinslerbundleasΩ0=−F.ThisbecomesΩ4=−FinthefifthdimensionoftheKKspacesincethee0oftheFinslerbundlebecomestheunitvectorutangenttothepropertimelines.KKspaceprovidesforarepresentationofelectromagnetismfreerfromspacetimeframesthanitsrepresentationinFinslergeometryis.ButthereismoretotheroleoftheK¨ahlercalculus.TheK¨ahlerviewofquantummechanicsthathesawemergefromhiscalcu-

24111.5.CARTAN-KAHLER-EINSTEIN-YANG-MILLS¨223lusmovesthefocusonthesymmetriesofbasicquantummechanicalequationstothesymmetriesthatcanbeputtogetherinsolutionsofthoseequations.Inotherwords,theidempotentsthatdefinetheidealsthatembodythosesymme-triescometothefore.FromtheansatzforthesolutionswithtimetranslationandrotationalsymmetryoftheDiracequationwithelectromagneticcouplingemergestheansatzforthesolutionsofequationswithU(1)×SU(2)symme-try.NoticeintheelectromagneticcasethatthosesolutionsdonotdirectlytheU(1)symmetry,butindirectlythroughitsentanglementwithproperenergy,andalmostasdirectlywithspin.Inotherwords,weareseeingintheansatzforsolutionswithsymmetrytherelationofU(1)×SU(2)tothePoincar´egroup.RegardingSU(3),SchmeikalhasshownthatthetangentCliffordalgebraofspacetimemayaccommodatequarksandthegeneratorsofSU(3)throughtheuseofmutuallyannullingprimitiveidempotents[66].BringinK¨ahler’streatmentofquantummechanicswithdifferentialforms—and,again,whathismethodssayaboutthesolutionofexteriorequationsforphysicalsystemsen-dowedwithtimetranslationandrotationalsymmetry.Thattreatmentprovidesinanaturalwayonesuchfamilyofmutuallyannullingprimitiveidempotents,butinaCliffordalgebraofdifferentialforms.Thatresultrequiresonlyscalar-valueddifferentialforms.Moregeneralvaluednesswillobviouslygeneratemanymorefamiliesofidempotentstoplaysimilarroles.InTP,wedonotneedtomakeU(1)×SU(2)×SU(3)consistentwith(pseudo)-Euclideancurvature—usuallybuterroneouslycalledaffinecurvature—sincealltangentspacestothebasemanifoldcanbeidentified.Theidealsatdifferentpointsofspacetimeor,morepointedly,space-propertime,canbesim-ilarlyidentified.TheauxiliarybundlesofYang-Millstheorymaynotbeneededafterall,sincetangentbundlegeometryinthesenseofourKKspacemayachievethesamegoals,butmorenaturally.Oneshouldbeabletoseehowthetwopartsoftheconnectionplaytheirrespectiveroleswhenonefurtherdevelopsthemanylinesofresearchthatthetheoryofmovingframeslaysinfrontofus.Andbeawareofthefactthatthefifthdimensionisnotsomethingtotallynew.Itisanewwayoflookingatpropertime.Itwasthereaslongasthereclocks,sayhydrogenatoms,andalmostaslongastherewastimeitself.

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243Chapter12UNDERSTANDTHEPASTTOIMAGINETHEFUTURE12.1IntroductionThefirstfourdecadesofthenineteenthcenturywitnessedtheworkofsuchmathematicalluminariesasGaussincalculusandgeometry,andCauchywithhiscomplexcalculusandhistheoryofalgebraickeys.Retrospectively,Iconsidertheirveryimportantworksthelastonesofapastera.Forthisauthor,anewerastartsin1844withthepublicationofHermannGrassmann’sOpusMagnum.TitledDieAusdehnungslehre,itisalsoknownastheCalculusofExtension.Butitismainlyaboutalgebra.Allthisthusspeaksoftheintimaterelationbetweenalgebra,calculusandgeometry.Grassmann’spublicationsareverydifficulttounderstand,buthavefortunatelybeeninterpreted.Intheopinionofthepresentauthor,theGrassmanneraisfarfrombeingover.Inordertounderstandwhy,weshallexplainthepresentstateoftheareasofalgebra,calculusandgeometrythatwere,directlyorindirectly,impactedbyhiswork.Wepresentabriefbiographyofthisextraordinarythinkerinoneoftheappendices.12.2Historyofsomegeometry-relatedalgebraInhispaper“ThetragedyofGrassmann”,distinguishedFrenchmathemati-cianDieudonn´eputGrassmann’smathematicalwork[43]inperspective[32].Hepointedoutthatin1844(tenyearsbeforeRiemann’sfamouslectureatG¨ottingenonthefoundationsofgeometry!),Grassmannwasdevelopingmathe-maticsforspacesofhigherdimensionandconstructingobjectsofdifferentgrades[43].Specifically,hewasdealingwithmultivectorsandallsortsofgeometricproductsatatimewhenthetheoryofvectorspacescomprisedonlyadditionof225

244226CHAPTER12.UNDERSTANDTHEPAST,IMAGINETHEFUTUREvectorsandtheirmultiplicationbyscalars.Ayearearlier,WilliamRowanHamiltonhaddiscoveredtheconceptofquaternions.Heproducedanalgebrawithaninterestingmultiplicationtablewiththreeindependentunitsofsquareminusone,andoneunitofsquareplusone,namelytherealunitnumber.Butonlyin1853tookplacehispublicationtitledLecturesonQuaternions,whichwastohimwhatDieAusdehnungslehrewastoGrassmann.In1873,WilliamKingdomCliffordstartedtopublishhisalgebraicworknowadaysknownasCliffordalgebra.Forhisscatteredcontributionstothissubjectsee[26].HegracefullyacknowledgedhisgreatdebttoGrassmann.Clifford’sworkallowsadeepcomparison,thoughindirect,oftheimpactsofthealgebraicworksofGrassmannandHamilton.ThealgebraofquaternionsisisomorphictotheCliffordalgebraCl2.Butitisjustonealgebra.Ontheotherhandandtosaytheleast,Grassmannproducedasystemofalgebras(theexterioralgebras)applicabletoanynumberofdimensions,andmuchmore.Seehisbiography.ButhewaslessspecificthanHamiltonwhenaquadraticformisinvolved.FollowersofGrassmannandHamiltonbecamebitteradversariesduringthenineteenthcentury.TheworkofGrassmannthusisincomparablefarmorerelevantthanHamil-ton’sandmorebroadthanClifford’s.AlthoughquadraticformsareoftheessenceinthegeometricworkofRiemann,thereisverylittlerelationofhisworktothatofCliffordandHamilton,whichwasalgebraicratherthandifferential-geometric.Itisimportanttobeawareofthefactthattheexteriorandinteriorprod-uctscanbeviewedasshorthandforcombinationsofCliffordproducts,aswealreadysawinchapter3.Hence,itistotallyunnecessarytocomplicatemattersbythinkingintermsofstructureswhereCliffordalgebrawouldbeasubstruc-ture,andexterioralgebrawithadotproductaddedtoitwouldbeanothersubstructure.ItseemstothisauthorthatthemainexampleoffurtherdevelopmentofGrassmann’sideasnothavingtodowithquadraticformshastodowithpro-jectivealgebra/geometry.Intheprojectiverealm,Peanofurtherdevelopedthoseideasin1888[58].SeetheinterestingopinionofGian-CarloRotaandcollaboratorsin1995ontheimpactofGrassmann,inthecontextoftheirownworkinthelineofthinkingofGrassmannandPeano[1].Followingcustom,weshallusethetermCliffordalgebrawhenitisbuiltuponatangentvectorspaceoratangentmodule.AndweshallusethetermK¨ahleralgebrawhenitisbuiltuponamoduleofdifferential1-forms[46],[48].ButweshallusethetermCliffordproductinbothcases.Thestateddevelopmentsinalgebraarenottheonlyonesintheepochthatweareconsidering.But,inourview,theyarethemostrelevantonesincon-nectionwiththeconvergenceofcalculusandgeometry.

24512.3.HISTORYOFCALCULUSANDDIFFERENTIALFORMS22712.3HistoryofmoderncalculusanddifferentialformsThereissomecalculusinthemistofGrassmann’salgebraicwork(Seeacou-pleofarticlesinelectrodynamicsandmechanicsinthecompilationofEnglishtranslationofmuchofhiswork[43]).Butgoingbeyondthisbriefstatementwouldonlydistractusfromhisincomparablymoreimportantalgebraicwork.SomethingsimilarbuttoalesserdegreecouldbesaidoftheworkofCliffordthattouchescalculus.OtherthanthecomplexvariablecalculusofCauchy,whichgoesbacktothebeginningofthenineteenthcentury,theonlycalculusofthatcenturythatisinheavyuseuptothisdayisthevectorcalculus.ItwasconceivedalmostsimultaneouslybythephysicistGibbsandtheengineerHeavisideinthe1880’s,usingideasfrombothGrassmannandHamilton.Wereferreaderstothebook“AHistoryofVectorAnalysis”byMichaelJ.Crowe[28],wheretheywillfindmoreempathyforGibbsandHeavisidethantheywillgetfromthepresentauthor.Alas,thereisinthevectorcalculusmoreofHamiltonthanofGrassmannsinceHamiltononlycaredaboutfourunits(1,i,j,k).Thencametheexteriorcalculusofdifferentialforms,proposedbyE.Cartan´in1899[4]inthecontextofsolvingsystemsofdifferentialequations.Itisunderlaidbyexterioralgebra,builtbyexteriorproductofcopiesofthemoduleofdifferential1−formsfordifferentiablemanifoldsofarbitrarydimension.Thisnewcalculuswasinfactmuchmorethanjustanewcalculusintheordinarysenseoftheword,asthatpapercontainedmaterialondifferentialformsthatisonlyseenintreatisesonexteriordifferentialsystems(anysystemofdifferentialequations,ordinaryorpartial,canbewrittenasanexteriorsystem).ThisisnotstrangesinceCartanwasstandingontheshouldersofsuchluminariesonthesubjectasLie,FrobeniusandPfaff.ThisworkbyCartanondifferentialformsconstitutedapaperwithinapaper.Itislikelythatitwouldhaveobtainedamuchfastertractionintheliteratureifithadbeenpublishedseparatelyfromtherestofthepaper,thusemphasizingtheappearanceofanewcalculus.Cartanwasvirtuallytheonlyonetousethistoolforafewdecades.Onedecisiveapplicationofhiscalculusisconstitutedbyhis1908paperonLiegroups[6].His1922book“LecturesonIntegralInvariants”[9]isamonumenttotheextraordinarypowerofexteriorscalar-valueddifferentialforms,speciallyinthefieldofclassicalmechanics.Asweshallintimate,onecanhaveamorecomprehensivealgebraofscalar-valueddifferentialforms(K¨ahler’s),and,correspondingly,astillbettercalculus.Outofrespectforunfortunatetradition,weshallusethetermdifferentialformswhentheyareconsideredaselementsoftheirexterioralgebra,ratherthanasmembersofmorecomprehensiveones.WhenthealgebrabeingusedisClifford’s(seesection3),weshallpreferentiallyrefertothemasclifforms.Inthemeantime,thetensorcalculusemergedfromacollectionofscatteredresultsspanningfourdecades[60].Itfilledagap,sincethevectorcalculusdoes

246228CHAPTER12.UNDERSTANDTHEPAST,IMAGINETHEFUTUREnotapplytoEuclideanspacesofarbitrarydimension,andtoRiemannianspacesofanydimensionexceptdimensionone.Thebestversionofthetraditionaltheoryofcurvesandsurfacesin3-DEuclideanspaceusedthevectorcalculus.Similarly,thetensorcalculusbecamethecalculusinRiemanniangeometry.In1934,ErichK¨ahlerproducedwhatisnowadaysknownastheCartan-K¨ahlertheoryofexteriorsystems[45].Fastforwardto1960-62.Inthoseyears,K¨ahlerpublishedhisworkonthecalculusthatbearshisname.Hecalledittheinteriorcalculus.Thenameexterior-interiorcalculuswouldbemoreap-propriate.K¨ahlerdevelopeditonlyformanifoldsendowedwithaRiemannianmetricandLevi-Civitaconnection,butthegeneralizationtootherconnectionsistrivial[78].K¨ahlerusedhiscalculusin1961toobtainthefinestructureofthehydrogenatomwithscalar-valuedclifforms[47].Moreimportantly,heusedthesetoshowhowtheconceptofspinoremergesinsolvingequationsofdifferentialforms,eveniftheequationsarenotrestrictedtotheidealstowhichthespinorsbelong[47].In1962[48],hepresentedthegeneraltheorymorecomprehensivelythanin1960[46],andusedittogetthefinestructuremoreexpeditiouslythanin1961[47].Averyimportantandyetoverlookedresultthatheobtainedisthatantiparticles(atleastinhishandlingoftheelectromagneticinteraction)emergewiththesamesignofenergyasparticles[48].Atitsmostgeneral,heconsideredtensor-valueddifferentialforms,forwhichhedidnotprovideapplications.Buteventherestrictionofhiscalculustoscalar-valueddifferentialformsproducesamazingresultswherespinemergesnotasaninternalpropertyofparticlesattachedtoorbitalangularmomentum,butatthesamelevelwithit.Amazingisthenaturalnessofhistreatmentofrelativisticquantummechan-icsthroughhisK¨ahler-Diracequationor,saidbetter,K¨ahlerequation,whichsupersedesDirac’s.Withthatequationandelectromagneticcoupling,antipar-ticlesemergewiththesamesignofenergyasparticles.TheauthorofthisbookhasshownthatcomputationsinrelativisticquantummechanicswithK¨ahler’scalculususingsimplyscalar-valuedclifformsaremucheasierandlesscontrivedthanwithDirac’scalculus[83].TheexpansionofthatequationusingthemassoftheelectronasthedominantenergytermgivesrisetotheclassicalformoftheelectromagneticHamiltonianasinthePauliequation.Inshortsuccession,thesameprocessgivesrisetotheHamiltonianatthelevelofthefinestructure,withoutresorttoFoldy-Wouthuysentransformations.Inretrospect,thesetransformationsarenotphysicallysignificant.TheK¨ahlercalculusofscalar-valueddifferentialformsistoquantumme-chanicswhatthecalculusofvector-valueddifferentialformsistodifferentialgeometry,andthustogeneralrelativity.Anobviousnextstepistofindappli-cationsoftheK¨ahler’scalculusinvolvingvaluednessotherthanscalar.

24712.4.HISTORYOFSTANDARDDIFFERENTIALGEOMETRY22912.4HistoryofstandarddifferentialGEOMETRYWeareconcernedwithdevelopmentsindifferentialgeometrythattookplaceafterthefirstpublication,in1844,ofGrassmann’sseminalworkonalgebra.ThusourhistoricalreportonthissubjectstartswithBernardRiemann’splant-ingin1854theseedsofwhatwouldlaterbecomethe“program”ofRiemanniangeometry.WehaveusedthetermprogramtoputRiemanniangeometryinthesamebagwithgeometrieswherethemainconceptistheconceptofdistance.Nothingmuchhappenedforadecade,whenRiemanntackledtheproblemofwhethertwoquadraticdifferentialformsarerelatedbyacoordinatetrans-formation.Itisaproblemofatypecalledproblemofequivalenceand,roughlyspeaking,thereisamethodtosolvesuchaproblemcalledmethodofequiva-lence.Itwasnotaverywelldefinedmethodatthetime,whichiswhywemayspeakofthedifferentversionsofitthat,accordingtoPauli[57],wereusedbyRiemann,ChristoffelandLipshitztoindependentlyobtainwhatisnowcalledRiemann’scurvature.Itsannulmentconstitutesanecessaryandsufficientcon-ditiontogiveapositiveanswertothespecificproblemgivenatthebeginningofthisparagraph.Fastforwardtotheyear1872,whenthemathematicianFelixKleinlaunchedhisfamousErlangenprogram.Wespeakaboutitatgreaterlengthinourear-lierbook“DifferentialFormsforCartan-KleinGeometry”,subtitled“ErlangenProgramwithMovingFrames”.Beforebuyingit,considerthatthecontentsofthatbookandthepresentoneislargelycommon.Butwehaveaddedtheappendices,totallyrenewedtheprefaceandchapters1,12and13,andaddedsectionsonCliffordalgebrainchapter4,aswellasafewscatteredsectionsinotherchapters.TheErlangenprogramwasbasedontheconceptofgroup.ItwasHenriPoincar´ewho,accordingtoElieCartan,interpretedKlein’sprogramasbeing´equivalentlyaboutgeometricequality[15].Inthemoderninterpretation—totheextentthatithasadoptedCartan’sviewsongeometry—Klein’sprogramistotheconceptofgeometricequalitywhatRiemann’sprogramistotheconceptofdistance.Kleinspaces/geometriesarenotnowadayswhatKleinhimselfconsideredasspaces/geometries.NoteveryKleinspaceofthattimeisaKleinspacetoday.Forexample,thesphereisnotthereceptacleofageometryinthemodernsenseofKleinspace.ThemodernKleinspacesare,sotospeak,flatspaces.CartanspacesaretheirmoderngeneralizationsinthedirectionthatCartanprovided.CartanspokeofaKleingeometryasthestudyofinvariantsunderthetrans-formationsofagroup.Lateinhislife,Cartanwroteexplicitlyabouttheroleofgrouptheoryinmoderndifferentialgeometry[21].HehadinmindpairsofagroupGandasubgroupG0,subjecttosomeconditionaboutwhichthereisnotunanimity.Buthedidnotneedtoelaborateonsomefurtherconditiontobesatisfiedsince,forthemostpart,hestudiedspecifictypesofgeneralizedgeometries,whichsatisfiedit,whicheverwethinkistherelevantone.

248230CHAPTER12.UNDERSTANDTHEPAST,IMAGINETHEFUTURECartanmentionedG0repeatedly,butonlyexceptionallydidhereferexplic-itlytothepairofgroupandsubgroup,althoughthisstructureisotherwiseveryclear,thoughonlyimplicit,inhiswritings.TheGgroupsarecalledthefundamentalgroupsofthegeometries.Onealsospeaksofthefundamentalgroupofthespacewherethestudyofthepropertiesofthefigurestakesplace.LetusrepeatCartan’swords[15]:Ineachofthesegeometriesandforexpediencyreasons,oneattributestothespacewherethefiguresunderstudyarelocatedtheverypropertiesofthecorrespondinggroup,orfundamentalgroup;onethusspeaksof“Euclideanspace”,“affinespace”,etc.,insteadof“spacewhereonestud-iesthepropertiesofthefiguresthatareleftinvariantbytheEuclideangroup,affinegroup,etc.”G0happenstobethesubgroupconstitutedbyalltheelementsofGthatleaveapointunchanged.AlltransformationsinG0arepresentinthefibersofthegeneralizedspaces.Thisisincontrastwithtransformationsliketranslations,whicharevalidonlyindifferentialforminthegeneralizations,andonlyinthebasemanifold.InEuclideangeometry,Gisthegroupofthesocalleddisplacements(trans-lationsandrotations,withthegroupofrotationsasG0);inaffinegeometryGisthegroupoftheaffinetransformations(translationsandlineartransfor-mations,withthelineartransformationsasG0);inprojectivegeometry,Gistheprojectivegroup(translationsandhomographies,withthehomographiesasG0).Theemergenceofthetensorcalculus,ofwhichwespokeintheprevioussection,maybeseenasthenextstepinthedevelopmentofdifferentialgeometry,exceptfortheadvanceintheformulationoftheclassicaltheoryofcurvesandsurfacesthathadtakenplacenotmuchearlierwiththeemergenceofthevectorcalculus.Incontrastwiththemethodofequivalence,thetensorcalculusalreadyhadadistinctivedifferential-geometricflavor.Duetothis,manyphysicistsofpastdecadeswouldviewthedifferentialgeometryofRiemannianspacesasvirtuallysynonymouswiththetensorcalculususedbythoseauthors.Suchwouldnotbe,however,thecase,forinstance,withthealgebraic,moreformalpresentationofthesamecalculusbyLichnerowicz[53].SignificantisthefactthatthederivationoftheexpressionfortheRieman-niancurvature(derivationlateradoptedbyEinsteininhisworkongeneralrel-ativity)hasmoreflavorofexteriorcalculus—disguisedasantisymmetrizationtoremovecertainterms—thanofmethodofequivalence.Levi-CivitawasdiscipleofRicci,butitistheformerwhogetsgreaterex-posureinCartan’swritingsbecauseoftheconnectionthatbearshisname.Itwasproposedin1917,andconstitutesthepointofcontactoftheKleinandRie-mannprogramsinthefollowingsense.UntiltheLevi-Civitaconnection,therewasnoconceptofgeometricequalityinRiemanniangeometry,i.e.ofequal-ityofvectorsatdifferentpoints.Tobeprecise,thereisnogeometricequalityintheLevi-Civitaconnectioneither,butwemayatleastspeakofgeometric

24912.4.HISTORYOFSTANDARDDIFFERENTIALGEOMETRY231equalityoftangentvectorsintheneighborhoodsoflines.Truegeometricequal-ityoverfiniteregionsofamanifoldonlyexistifthemanifoldisendowedwithteleparallelism.In1922,E.CartanpublishedaseriesofNotestotheFrenchAcademieof´Sciencesannouncingwhatwastobecomeoverseveralyearsaseriesofpapersonhistheoryofmovingframes.ThisisatheorywhichappliesinprincipletoallKleingeometries.WehavedealtinthisbookwiththeaffineandEu-clideancases,bothelementaryandgeneralized.CartanunifiedtheRiemannandKleinviewsofgeometrythroughageneralizationofKlein’sErlangenpro-gramof1872.Dieudonn´espeaksofthisgeneralizationasfollows(reproducedfromtheintroductionofabookbyGardner[42]):“...itisfittingtomentionthemostunexpectedextensionofKlein’sideasindifferentialgeometry.HehadenvisagedgroupsofisometriesinRiemannianspacesasapossiblefieldofstudyofhisprogram,butingeneralaRiemannianspacedoesnotadmitanyisometriesexceptfortheidentitytransformation.Byanextremelyoriginalgeneralization,E.Cartanwasabletoshowthathereaswelltheideaof“operation”´stillplaysafundamentalrole;butitisnecessarytoreplacethegroupwithamorecomplexobject,calledthe“principalfiberspace”;onecanroughlyrepresentitasafamilyofisomorphicgroups,parameterizedbythedifferentpointsunderconsideration;...”Regrettably,Cartan’sresultsarepresentednowadaysinwaystotallydiffer-entfromhisoriginalone.ButthesemodernwaysdonotexhibittheessenceofCartan’sprogram,howhisworkgeneralizesKlein’s,andhow,intheprocess,Riemann’sprogramisabsorbedintoit.Aswehavemadeabundantlyclear,CartanapproachedhisgeneralizationofKlein’sgeometriesasaproblemofintegrability,whoselanguageisthelan-guageofdifferentialforms.WhenCartandealtwithwhatothersconsidertobetensor-valuedness,hewasactuallydealingwithexteriorproductsoftangentvectorbasesoroffieldsthereof.Somethingthatlookslikeaskew-symmetric(tangent)tensor,neednotbeatensor.Itmaybeamultivector.Thekeyastowhetheroneisdealingwithoneortheotherlieswiththeproductsanddifferentiationsthatweperformwithandonthem.Sufficetosaythatthetensorproductofskew-symmetrictensorsdoesnotyieldingeneralanotherskew-symmetricone.Cartaninformallyintroduced,inadditiontotheexterioralgebraofdifferen-tialforms,anexterioralgebraofvaluedness.Heextendedtoittheactionofhisdoperatorofdifferentiation,whichisthereasonwhyhe(andK¨ahler)usethetermexteriordifferentiationwhenothersusethetermexteriorcovariantdiffer-entiation.ReaderswhowouldlikeanapproachtothisextensionmoreformalthaninthisbookshouldconsultFlanders[41].NextinlinewouldbeCartan’sworkonFinslergeometry.Wemustbackpedalalittlebitto1918.ItwasPaulFinslerwho,undertheguidanceofhisdoc-toralsupervisorConstantinCarath´eodory,studiedthegeometryofcurvesand

250232CHAPTER12.UNDERSTANDTHEPAST,IMAGINETHEFUTUREsurfaceswhereaconceptofmetricmoregeneralthanRiemann’swasdefined(ThisconcepthadbeenbrieflybroachedbyRiemannhimself).ButthiswasageneralizationoftheoldRiemanniangeometry,sotospeak,i.e.onewithoutconnections.ThusthewideusethathasbeenmadeofthenameofFinslergiveshimimplicitlyfarmorecreditthanhedeserved.InfactthefirstEuclideanconnection(i.e.metriccompatibleaffineconnection)onaFinslermetricisduetoCartanin1934[19].ItisknownastheCartan-Finslerconnection.ItistoFinslermetricswhattheLevi-CivitaconnectionistoRiemannianmetrics.ItistroublesomethatmodernFinslergeometersworkinwhatamountstosectionsofabundlebutgenerallyfailtodefinewhatthebundleis.Iftheydid,theywouldhavetoconsiderwhatareaffine-Finslerframebundles,andwhattherestrictionofsuchbundlesthatonecouldcallmetric-Finslerbundlesis.NotevenCartandidexplicitlydealwiththisissue.Considerthestandardspacetimeofspecialrelativity.WecanpushittoitsFinslerbundle.Inotherwords,wecanreorganizetheframesoveritsspherebundleorbundleofdirections.Asetofcoordinatesinthebasespacecouldbe(t,xi,uj),theuj’sbeingvelocitycoordinates.Thegroupinthefibersisthegroupofrotationsin3-D.WeshallnameitG00.Wethushavethetriple(G,G0,G00).G0(Lorentzgroup)nowplaysadiminishedyetstillrelevantrole.Itssignificancecanbeclearlyascertainedindealingwithalternativestospecialrelativity.Itisthebridgetotheframesofrelevanceinthosealternatives.Inonewayoranother,therearethreegroupsinthegeometryofspecialrelativitywhenviewedfromaFinslerbundleperspective:thePoincar´egroup,itsstandardLorentzsubgroup,andthelatter’slargestsubgroupofrotations,O(3).Butthishasnotbeenformalizedand,inanycase,transcendsthescopeofthepresentbook.BeforewecontinuewithFinslergeometry,consideramid-centurydevelop-mentofsignificancefordifferentialgeometry.ItisthealreadymentionedK¨ahlercalculus.Itisofspecialimportanceforquantumphysics,butitdoesnotimpedethat,intheprocess,thebasicmagnitudesofdifferentialgeometryemergeevenwithoutresorttoissuesofgeometricnature.Hedisposedofissueslikecovariantdifferentiationbyansatz,theonlyconnectionheconsideredbeingtheLCC.Hedidnotdealwithbundles,norwithwheredoconnectionstaketheirvalues.But,tosaytheleast,differentialgeometryandquantummechanicsnowshareinhisdifferentialformsacommonmathematicallanguage.Hence,hecontributedtothedevelopmentofgeometry,thoughtheimplicationsofhisworkarenotyetseen.ArelativelylittleknownbutverysignificantdifferentialtopologistbythenameofYeatonH.Cliftonentersthepicturenow(Thefamousglobaldifferen-tialgeometerS.-S.CherntoldthisauthorofhisgreatrespectforCliftonasamathematician).In1989,ImadeprogresswiththeissueofgeometrizingtheequationofmotionofspecialrelativitywithLorentzforceinatangentbundlecontext.TheconnectionhadtobeviewedasFinsleriantomakesense,thoughIhadobtaineditwithoutexplicitlyresortingtoFinslergeometry.Icommu-nicatedthisresulttoClifton.Itgreatlyimpressedhimbecausehehadtriedhimselfseveralclassical-geometricoptionswithoutfindinganythingthatsatis-

25112.5.EMERGINGUNIFICATIONOFCALCULUSANDGEOMETRY233fiedhim.Ithusearnedhisrespectandwestartedacollaborationwhereheprovidedgenialresponsetomyquestions.Mymainquestionwas:ifEuclideanconnections(i.e.whatareusuallycalledmetriccompatibleaffineconnections)arerestrictionsofaffineconnectionsbyvirtueofrestrictionofthebundlesonwhichaffineconnectionslive,ofwhataffineconnectionsaretheusualFinslerconnections(i.e.basedonmoregeneralconceptsofdistance)restrictions?Afterrecognizingthathehadnotaskedhimselfthatquestion,heimmediatelyprovidedaseriesofgreatresults.Amainonewasatheoryofaffine-Finslerbundles,i.e.notrequiringametricstructure.Inparticular,suchconnectionscouldberestrictedbytheLorentzmetricstructure,whichis(pseudo-)Riemannian.Insuchastructure,onecanaccommodateamagnitudesuchastheelectromagneticfield,whosecomponentshavetwoindices,withinthetorsion,whichhasthree.OnereallyappreciatestherelevanceofClifton’sworkwhenonestudiesthemodernpresentationsofdifferentialgeometry,speciallythetheoryofconnec-tions,andthenreadsthepapersbyCartanwherethattheoryispresentedforthefirsttime.Onewouldthinkthatoneisdealingwithtwodifferenttheo-ries.ThemodernpresentationsappeartohavebeenmotivatedbythefactthatCartan’sworkisoftenconsiderednottoberigorous.Cliftonmadeitrigorousthroughafewdefinitionsandastillsmallernumberoftheorems.Theproblemofspecializingthattothesimplepre-FinslerianaffineandEuclideanconnectiontomakethemrigorouswasthenasimpleexercisethatItookcareof.Wehavedealtwiththis“specialcase”onitsowninsection13ofchapter8,andsection2ofchapter12.TosummarizethesignificanceofClifton,letussaythattheunderstandingthathehadofCartan’sworkpermittedhimtoformulateveryelegantlyatheoryofaffineFinslerbundles,i.e.Finslergeometrywithoutaconceptofdistance.ItisjustasimpleextensionofthetheoryofaffineandEuclideanconnections.Asaby-product,wenowalsohavearigorousformulationforthetheoryofaffineandEuclideanconnectionswiththemovingfamemethod,whichwasthoughttobeimpossible,atleastifonedoesnotchangeitsoriginalflavor.Finally,letmeaskaquestionforhonestdifferentialgeometers.CantheyformulateatheoryofconnectionsonFinslerbundleswiththemodernmethods?12.5EmergingunificationofcalculusandgeometryInthetensorcalculus,onlythelowestranksoftensor-valuednessemerge.Thisseemstoindicatethatgeneraltensoralgebra,whichisinfinitedimensional,isonlyindirectlyrelevant,asmotheroffinitedimensionalquotientalgebras:exterior,Clifford,....Thereareotherreasonstoignoretensorvaluedness,whichwillbepresentedinthenextchapter.TheyarethealgebrasoftheCartanandK¨ahlercalculi.Thelatterauthordidnot,however,produceappli-cationsevenforvectorvaluedness.Hisscalar-valuedapplicationsare,however,

252234CHAPTER12.UNDERSTANDTHEPAST,IMAGINETHEFUTUREofgreatimportanceforthedevelopmentofphysics,asweshallseeinthenextsection.Geometry,ontheotherhand,isbasedontheconnectionequations,dP=ωμe,de=ωνe(5.1)μμμν(Seesection5.7).ThealgebrahereistheLiealgebra,whichiswheretheconnec-tiontakesitsvalues.Morerelevantforpresentpurposesisthatwearedealingherewithvectorvaluednessandexterior(covariant)differentiation.Cartan’sdifferentialgeometrymightbeextendedifonefoundhowtogetinteriordif-ferentiationintoitinanaturalway.Itwasnotnecessaryinourdevelopment.Somenewconceptwouldbenecessarytomakeitsintroductionnatural.Forthemoment,itwouldappearthatanapproachbetweencalculusandgeometrymustcomefromthesideoftheformer.ThekeyequationoftheK¨ahlercalculusis[46]∂u=a∨u.(5.2)Whenaistheappropriatescalar-valueddifferentialform,itreplacestheDiracequationwithelectromagneticcoupling.Hedidnotspeakin1960ofacorre-spondingequationfortensor-valuedinputa.Theequation∂u=a∨u,(5.3)wheretheCliffordalgebrareferstothedifferentialformsthemselves(K¨ahleralgebra),remainsvalidregardlessofvaluedness.Whenaisnotascalarfunction,ucannotbeofhomogeneousvaluedness(saytensorvaluednessofdefiniterank)sinceitincreasestherankoftherighthandside,which∂,onthelefthandside,doesnot.Thevaluednessoftherightandlefthandsidesof(5.3)wouldnotbethesameexceptforinhomogeneousvaluednessextendingallthewaytoinfinity.K¨ahlerdealtwithtensor-valuednessainhis1962paperbyintroducingacontrivedK¨ahlerequationthatreads(∂u)λ...π=aλ...πp...gur...s.(5.4)μ...νμ...νr...sp...gWedonotneedtolearntheconceptsthatgointothemakingofEq.(5.4).Sufficetonoticethattherankofthevaluedness-tensorinputaλ...πp...gisdoubleμ...νr...stherankofur...s.Equation(5.2)isnotsorestricted.Bothofthemthushavep...gtheirownun-invitingfeatures.K¨ahlerstatedin1962that(5.4)becomes(5.2)inthespecificcaseofscalar-valuedness.Thatisatruebutmisleadingstatementsincehecouldhavepos-tulated(5.3),thusavoiding(5.4).Hecouldthenhavesaidthatbothequations(5.3)and(5.4)coincideforscalar-valuedinput.Ratherthanconcernourselveswiththepossiblereason(s)behindK¨ahler’s1962choice(orlackthereof),itismoreimportanttoconsiderwhatpost-scalarvaluednessshouldaphysicistbeinterestedin.AnotablealternativeisCliffordvaluedness,aswehavemadeabundantlyclearin[88],whereitemergesnaturally,asshowninapreprintthatwesub-mittedtothegeneralphysicssectionofthearXiv[89](Itwasredirectedtothe

25312.6.IMAGININGTHEFUTURE235theorysectorofhighenergyphysics(wherewedonothaveasponsor).TheCliffordvaluednessofdifferentialformsthereemergesnaturally.WenowhaveCliffordvalueddifferentialformsastheactorsforthepotentialunificationofcalculusandgeometry.Thispotentialitynowhastobecomerealizedandthusbringuscloselyinprincipletotheunificationofquantummechanicsandgeneralrelativity.12.6ImaginingthefutureThehistoriesofmathematicsandphysicsarefullofaccidentsthathavemadethemhavetheirpresentshapes,fullofavoidabledeficiencies.Replacingthemistheroadmaptothefuture.Thereplacementsexist,someofthemoverlookedformorethanacentury.Byaccidentsinmathematics,wemeanthatthehistoricalorderofmathemat-icaldiscoverydidnotcorrespondtowhat,retrospectively,couldbeconsideredasthelogicalorder.ExamplesareCauchy’scalculus,largelyortotallyunnec-essary,aswehaverecentlyshown.Ortheadoptionofvectoralgebra,whichispeculiarbecauseitisspecifictothreedimensions.OrtheviewofthegeometryofLorentzianspacesaspseudo-RiemannianratherthanFinslerian(becauseofgivingmoreimportancetothemetricthantothestructureofthebundle).OrthedisconnectofdifferentialgeometryfromthelanguageofphysicistsduetotheattempttomakeCartanrigorousthroughtheuseofforceps.Theaccidentsinphysicsarelargelyaconsequenceoftheaccidentsinmath-ematics,whenthenotsogoodwaschosenbecausethebetterdidnotyetexist.AnexampleistheadoptionoftheLCconnectionatatimewhenitwastheonlyconnectionknown.AnotherexampleistheDiraccalculus,adoptedwhentheK¨ahlercalculuswasnotyetknown.Thoseaccidentsneednothaveconstitutedaproblemexceptforthefactthatinertiaandthelackofinterestinthefoundationsofphysicsand/ormathematicsimpededtheadoptionoftherightoptionwheniteventuallybecameavailable.Assumethatdifferentialgeometershadfocussedtheirinterestmoreonfur-therdevelopingthemovingframemethodthanonmakingitrigorousincon-trivedways.Theymighthaverealizedthatmoderndifferentialgeometryis—regardlessofwhowedressit—atheoryofjustmovingframes.Itshouldbeatheorywhereparticlesandfieldsshouldplayamorecentralrole.Afterall,framesareforreferringparticlesandfieldstothem.Ihaverecentlyshownthat,whenthisisdone,a5-DKaluza-Kleinspacewith-outcompactificationofthefifthdimensionemerges[89].Thefifthdimension,τ,isonewhich,ontrajectoriesofparticles,becomestheirpropertime.The(xi,τ)subspaceisthenaturalarenaforquantumphysics.U(1)×SU(2)isthenshowntoappearbysimplyusingK¨ahler’scalculusinrelativisticquantummechanicsandreplacingtheunitimaginarywithappropriaterealmagnitudesofsquareminusone[89].Inworkinpreparation,U(1)×SU(2)×SU(3)willbeshowntobejustacontinuationoftheprocessbywhichU(1)becomesU(1)×SU(2).Wegivetheunderlyingideainthelastsectionofthenextchapter.

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255Chapter13ABOOKOFFAREWELLS13.1IntroductionIntheprevioustwochapters,wehavemadethepointthatthereisalargeamountofunfinishedworkinthelinesofgeometryandcalculuscreatedbyCartanandK¨ahler.Inthischapter,wemakethepresentationofournextbook,wherenewlinesofmathematicaldevelopmentareoutlinedwhennotdeveloped.Howcanphysicsmakeprogresswithitsmostdifficultissueswhileignoringsuchrelatedmathematics?Thecasewetrytomakeinitiscontainedinthetitle“Differentialforms:farewelltovector,tensor,CauchyandDiraccalculi”.Inordertobeabletobidfarewelltothem,wemusthaveareplacement.WeshowthattheK¨ahler’scalculusofdifferentialformsnotonlyreplacesbutactuallysupersedesthem.13.2FarewelltovectoralgebraandcalculusPractitionersofCliffordalgebraknowthatitreplacesvectoralgebrawithad-vantage.Inthissection,wesummarizeourattempttomakethatpointinournextbook.SincethesubjectofthatbookisnotCliffordalgebraperse,itstreatmentthereofthisalgebrawillbeonlyascomprehensiveasneededforre-placingvectoralgebra,andforotherobjectivesofthisbook,likereplacingthevectorandCauchycalculi.Cliffordalgebraisnotanalgebracreatedbymathematicianstomakephysi-cists’lifemiserable.IthastobeconsideredasthetrueEuclideanalgebra,validinarbitrarydimension.Standardvectoralgebraisvalidonlyindimension3.Thereisnotasimilaralgebrainotherdimensions,thoughthereisavectorprod-uctinsevendimensions.Butitisadifferenttypeofalgebrasince,perpendiculartotheplaneoftwovectors,thereisawholefivedimensionalspace.WheneveratleastoneoftwofactorsinaCliffordmultiplicationisofgradeone,thedotand,aftersurgery,vectorproductscometogether.Whysurgery?Thevectorproducta×bisthecompositionoftheexteriorproduct,a∧b,with237

256238CHAPTER13.ABOOKOFFAREWELLSwhichwehavealreadybecomefamiliarintermsofdifferentialforms,andHodgeduality(Seesection6.4).Thesurgeryconsistsinnottakingthedualofa∧b.Vectorcalculusishorribleforseveralreasons.Oneofthemisthatitscurlisbasedonthevectorproduct.So,wedonothaveacurlinotherdimensions.Anotherreasonisthatitusestangentvectorswhereitshouldusedifferentialforms.Onemoreisthatitoftenusesmorestructurethanneededtosolveaproblem,sayametricstructure.Stillanotheroneisthatonecandosolittlewithitthatithastobecomplementedwithalltheothercalculithatwealsothinkofreplacingwithdifferentialforms.TheK¨ahlercalculus—basedonClif-fordalgebraofdifferentialforms—replacestangentvectorsandtangent-valuedoperatorswithdifferentialformswhosecoefficientsarerespectivelyfunctionsandoperators.Thereplacementofmuchofthevectorcalculuswiththecalculusofdiffer-entialformsisnothingnew.Itcanbefoundindifferentbooksontheexteriorcalculus,sayforobtainingcurlanddivergenceincurvilinearcoordinates.IntheK¨ahlercalculus,theyaregivenbytheexterior(dα)andinterior(δα)derivativesofadifferential1−form,notofavectorfield.Thetwooperationsarejustpartofone,∂(=d+δ)(Seeequation(5.1)ofthepreviouschapter).Itiseasytocheckthat,ifαisadifferential1−forminEuclidean3-space,dαisadifferential2−formwiththesamecomponentsasthecurlofthevectorfieldthathasthesamecomponentsasα.Andδαisthescalarknownasthedivergenceofthatvectorfield.Theyaremucheasiertoworkwith,actuallywithoutresorttoHodgeduality.Inaddition,∂actsondifferentialformsofanygradeandofanyvaluedness.K¨ahlerdifferentiationalsoappliestovectorfields,butnottoyieldthecurlandthedivergenceofavectorfieldinthevectorcalculus.K¨ahlerdefinestheinterior“derivative”intermsoftheLCConthedifferen-tiablemanifold.Hethenshowsthathisdefinitioncoincideswhatinthemodernliteratureisdefinedastheco-derivative,whichisobtainedintermsofthemetric.K¨ahler’sdefinition,however,maybeusedwithanyEuclideanconnection.Neither∂norδsatisfythe(presentformofthe)Leibnizrule,meaningthattheyarenotderivativesinthemodernsenseoftheword.However,followingK¨ahler,weshallstillusethetermderivative.Itisonlyamatterof“relaxingtheconceptofLeibnizrule”,asithadbeenrelaxedinthepastwiththeemergenceoftheexteriorderivative.Theforegoingconsiderationsmakeacaseforfarewelltovectorcalculus,afterfirstgettingaglimpseofwhatitis.Letitbeclearlyunderstoodthatwedonotadvocatenottoteachthevectorcalculus,butthatonedosoonlytothepointwherethetransitiontotheexteriorcalculusiseasiest.Inanycase,duetotherathergeneralizeddeteriorationinacademicstandards,onedoesnotgetmuchmorethanaglimpseofvectorcalculusinmostcoursesonthesubject.Inaddition,thatknowledgecanberecycled.

25713.3.FAREWELLTOCALCULUSOFCOMPLEXVARIABLE23913.3FarewelltocalculusofcomplexVARIABLEThealternativetotheCauchycalculusthatweproposeisadirectapplicationoftheCliffordalgebraofdifferentialformsintheplane.WegivethedefiningrelationsbothinCartesianandpolarcoordinates:22221dx=dy=dρ=1,dφ=.(4.1)ρ2Hence,2dxdy=−dydx,(dxdy)=−1.(4.2)Asfarasalgebraisconcerned,thecomplexlookinginhomogeneousdifferen-tialformz≡x+ydxdy,(4.3)playsintherealplanetherolethatthecomplexvariablex+iyplaysinthecomplexplane.Itemergesfromtherelationbetweendφanddy:xdy−ydxx−ydxdy11dφ==dy=dy=dy,(4.4)x2+y2x2+y2x+ydxdyzwherewehaveusedthat(x+ydxdy)(x−ydxdy)=x2+y2,(4.5)byvirtueofthesecondequation4.2.Weproceedtoprovideanoutlineoftheargumentforthetheoremofresidues.∞GiventheintegralF(x)dx,considerthedifferential1−form−∞α≡F(x+ydxdy)dx(4.6)onaclosedcurveformedwiththexaxisoftherealplaneandasemicircleΓatinfinity.Ifα−→0astheradiusofthesemicirclebecomesgreaterandΓgreater,wemaythenwrite∞F(x)dx=α.(4.7)−∞Giventhepull-backofαtopolarcoordinates(asinthephysicsliterature,weignorethepull-backsymbol),α=h(ρ,φ)dρ+j(ρ,φ)dφ,(4.8)jis221j=ρ(α·dφ)=ρ[F(x+ydxdy)dx]·(dy),(4.9)zintermsofCartesiancoordinates.Letususethesymbol“∗”toindicatereversionoftheorderofdxanddy,whichchangesthesignofdxdy,aswhenreplacingiwith−iincomplex

258240CHAPTER13.ABOOKOFFAREWELLSconjugation.WithFstandingforF(z),therighthandsideof(4.9)furtherbecomesρ211∗∗j=Fdxdy+F(dxdy)=(Fzdxdy)0=(−Fz)2,(4.10)2z∗zwherethesubscriptszeroandtwopickupthescalarpartandthecoefficientofthe2−formpartrespectively.ThelastoneisequivalenttopickingtheimaginarypartofF(z)z.Letustakethenextstepinasimplifiedway.Ifthecontourintegralhasapoleoffirstorder,onegetstheintegrationcirclegotozeroand,underappropriateconditionsthatweshallignorehere,onepulls("−Fz)2outoftheintegral,whichthenbecomesjustdφand,therefore,2π.Thewellknowntheoremofresiduesasitappliestofirstorderpolesresults.Ifthepoleisofhigherorder,oneproceedsasinCauchy’stheory.Asifthatwerenotenough,theadditionaltheoreticaldevelopmentsnecessaryforspecialpurposesintheCauchycalculusnowbecomemuchsimpler.Itshouldnotbesur-prisingsince,again,differentialformsconstitutethelanguageofintegration.Inparticular,theequivalenceoftheconceptsofanalytic(powerseriesexpansion),holomorphic(existenceofderivatives)andtheCauchy-Riemannconditionsbe-comemuchmoretransparentsinceonedoesnotneedtocreateaconceptofdifferentiationwithrespecttothezoftheCauchycalculusortothenewz.Differentiationwithrespecttorealvariablessuffices.Retrospectively,differentiationsandintegrationswithrespecttoacomplexvariableconstitutealessthanidealtreatmentofproblemswhosesolutionsshouldbedealtwiththelanguageofdifferentialforms.RealintegralspresentlyperformedwithCauchy’scalculusshouldjustbeviewedwiththerightperspec-tive,namelyasanextensionofacorollaryofStokestheoremoftherealcalculusinregionsoftherealplanethatarenotsimplyconnected.Twopapersbythisauthordealwiththissubjectrathercomprehensibly[86],[87].Thesecondoftheseextendsthetheoryjustdescribed,soastodealalsowithintegralswheretheresultisofthetypeu+vdxdy.Theyareaccessibletoeverybodyfreeofcharge.Ournextbookshouldpickupthecontentsofthosetwopapersandprovidenumerousadditionalworkedexamples,sothatonecanteachinafarshorterandmoretransparentwayintherealplanewhatnowadaysgoesbythenameofcalculusofcomplexvariable.13.4FarewelltoDirac’sCALCULUSDirac’scalculusandtheorywasafantasticachievementofthearguablysecondgreatestphysicistofthetwentiethcentury;hisworkmayhavehadgreaterim-pactthanevenEinstein’s.Onecouldhardlyhavedonebetteratthetime.Butthatwasthen.WhencomparedwithK¨ahler’scalculus,bornafewdecadeslater,Dirac’sleavesalottobedesired.Thatwillbeshowninachapterinournextbook,wherethemainpurposewillbetoshowthattheadvantageofK¨ahler’scalculusliesinitsuseofdifferentialformswhereonenowusesgammamatrices.

25913.4.FAREWELLTODIRAC’SCALCULUS241OfspecialinterestisthetreatmentoftheLiederivativeofthoseformsfromtheoriginalandoverlookedperspectiveofthetreatmentsofLiedifferentiationofthesubjectbyCartan[9],K¨ahler[46],[48]andSlebodzinski[69].TheextensiveusemadeofDirac’scalculushastaughtphysiciststonavigateitswaters,butsomedifficultspotsremain.Wespeciallymeanspuriousnegativeenergy.ThroughhisreplacementoftheDiracequationwithhisownequation,K¨ahlershowedtheemergenceofantiparticleswithoutresorttonegativeenergysolutions[48].Healsoshowedthatthereisnothingmysteriousaboutspin.Asheputit[48]:Thespinoftheelectronwillbeinterpretedasthenecessitytorepresentthestateofanelectronbyastatedifferentialratherthanbyastatefunction.ItisnottheintentionofthisauthortoreproduceinhisnextbooktheK¨ahlercalculusinallitsglory,butenoughofitinordertoshowbycomparisonhowunnecessarilycumbersome,andmisleadingatcertainpoints,theDiractheorycanbe.Weproceedtogivesomeexamplesadditionaltotheonesjustmentioned.Thisauthorhasshownthattheobtainingofthe“postPauli-Dirac”approxima-tionoftheK¨ahlerequationwithelectromagneticcouplingislittlemorethanaonepagedevelopment,halfofittakenbythestatementofresults.ItisdonewithoutresorttoFoldy-Wouthuysentransformationsandwithoutthespuriousinterpretationofsmallcomponentsasrepresentingnegativeenergycontribu-tions[83].NeitherFoldy-Wouthuysen,noranybodyelseforthatmatter,ap-pearstohaveaddressedthefactthatthesamedevelopmentoftheequationshouldapplytoantiparticles,exceptforthesignofthecharge.AtthesametimeasoneshowsthattheFoldy-Wouthuysenworkismis-leading,oneinterpretsthe−mc2termastheresultofaddingaspuriousterm−2mc2tothetruerestenergymc2[83].Onecannotignoretheproblemofneg-ativeenergiesintheDiractheoryonthegroundsthatitdoesnotariseintheverysuccessfulquantumfieldtheory,sincethelatterprovidescorrections,butnotthebasicvaluesthatitcorrects.Relatedtotheforegoingquotation,K¨ahlercouldhavebeenmorespecificandstatedinthesamebreaththattheactionoftheLiederivativeondifferentialformsisattherootoftheemergenceofspin.IfhehadraisedmoreforcefullyandintheEnglishlanguagethisissue,hemighthavelaunchedareassessmentoftheuseoftheDiractheoryinhighenergyphysics.Inparticular,hewouldhavepromotedEinstein’sviewofparticlesasconcentrationsofthefieldThewayinwhichonetinkerswiththeDiracequation(perhapsunavoidablybecauseoflimitedsophisticationoftheDiraccalculus)forobtainingfreeparticlesolutionsisattherootofthespuriousemergenceof(particle,antiparticle)pairswithoppositesignoftheenergy.K¨ahlershouldhaveemphasizedthatthesolutionstohisdifferentialequationareinprinciplemembersofhisalgebra,andnotonlymembersofthosealgebras’ideals.Thespinorsolutionsareonlyspecificsolutions,notgeneralones.Akeyissueandsourceofinterestingquestionsis

260242CHAPTER13.ABOOKOFFAREWELLSwhatsymmetriescanbeputtogetherinthesameideal,giventheidiosyncrasiesofproductsofidempotentscorrespondingtodifferentsymmetries.Alittlebitofthiscanbeseeninsection6.Tosummarize,Dirac’stheorymaybeconceptuallywronginspiteofthetremendoussuccessofthehighenergytheorybuiltuponit.SolutionsoftheK¨ahlerequationpertainingtothealgebraofclifformsmaybetheprimordialconcept;thesolutionsinideals,thusrepresentationsofparticlesasspinors,arederivedconcepts.Oneby-productisthatalsoquarksareaderivedconceptbyamechanismsimilartotheonebywhichparticlesemergefromtheprimordialfield.Thesestatementsaresupportedinpapersbeingprepared.Itisthein-tentionofthisauthortoputintothebookthatweareannouncingherethemathematicsneededforunderstandingtheseclaims.13.5FarewelltotensorcalculusForcompletenesspurposes,achapterofthebookbeenannouncedwillprovideasummaryofthemainpartofthepresentbook.Itspurposeistoexplainwhyoneshouldbidfarewelltothetensorcalculusandtoapproachestodifferentialgeometrythatdrawsignificantlyfromit.Itwillemphasizethepointsabouttobementionedinthissectionandshouldputtogethersomeoftheperspectivesthatnon-expertreadersshouldhaveacquiredfromreadingthepresentbook.Tensorcalculuslackstruegeometricpedigree.Indeed,tensorfieldsrelatetosectionsoftheframebundle,nottotheframebundleitself.Buttheframebundle,notthesections,representthegroupsthatdefineaKleingeometry.Tensorcalculusemphasizesasgeometricwhatisnot,apointmadebyCartanwhenhearguedagainstviewingRiemanniangeometryasthegeometryofanin-finiteLiegroup[21].Yang-MillstheoryinvolvesconnectionsvaluedinfiniteLiealgebrasofauxiliarybundles.Connectionsinnon-Yang-Millsdifferentialgeom-etryalsotakevaluesinfiniteLiealgebras.Thisshouldbeemphasizedinordertoavoidbeingmisleadintoviewingbytheinfinitegroupofdiffeomorphismsasunderlyingthegeometry.Tensorcalculusbeliesthenatureoftheobjectsthatittriestorepresent.Tensorsbelongtoagradedalgebra,butonlythelowestranksoftensorsemergeingeometry.Rankshigherthanfivedonotappear.Tensoralgebrausurpstherolesofitsquotientalgebras.Tensorcalculussubvertsthelogicalorderofconcepts.Itsmostcommonver-sionmakesverylittleemphasisontensoralgebraandfocussesonhowthingstransform.Butthetransformationpropertiesareaconsequenceoftheunder-lyingtensoralgebra,whichmostauthorsforphysicistsbarelydefine.Asaconsequence,resultshappeninahaphazardway.Suchisthecaseforinstancewiththeargumentto“discover”curvatureintheapproachofRicciandLevi-CivitatoRiemanniangeometry,approachthatEinsteinadopted.Curva-turethereemergesnotasaconcept,butasasetofsymbolsthattransforminaparticularway.

26113.6.FAREWELLTOAUXILIARYBUNDLES?243Onemorerespectinwhichtensorcalculusfailstomakejusticetothefounda-tionsofgeometryisitscasualtreatmentofissuesofintegrabilityandequationsofstructure,speciallyofthefirstequationofstructure.Itthusproducesthewrongstatementthatthetorsionistheskew-symmetricpartoftheconnection,whichiscorrectonlyincoordinatebases.Tensorcalculusistoocoarse,asitlacksdiscriminatingpower.Itfailstodis-tinguishthedifferentnatureofobjectswhosecomponentstransforminthesameway.Asaresult,onespeaksofdifferentdifferentiations(covariant,exterior,...)ofamathematicalobjectlike,forexample,wi.Tensorcalculusistoorigid.ItmakesFinslergeometrytoocumbersome,dueinparttothefactthatthedimensionofthespaceoftangentvectorsandofthemoduleofdifferential1−formsaredifferentinthatgeometry.TensorcalculusisinimicaltoKleingeometries,bundlesandLiealgebras,nottomentionvectorequality.Tensorcalculusisonewhosetimeislongpast.13.6FarewelltoauxiliaryBUNDLES?Thisauthorclaimsthatauxiliarybundlesareunnecessaryifonesimplyusestherightmathematics.ThekeyliesinrecognizingtheimplicationsofviewingquantummechanicsfromtheperspectiveofK¨ahler’sratherthanDirac’scalcu-lus.Anewvisionofquantummechanicsemerges.First,letusmakethecasefortheK¨ahlercalculusandconcomitantquantumtheory.1.ThiscalculusisthenaturalextensionoftheCartancalculus,asittakesintoaccountnotonlyexteriordifferentiation(curlsifyouwill)butalsointeriordifferentiation(divergences)withdifferentialforms.2.ItismorecomprehensivethantheDiraccalculussincethespinors(eightrealcomponents)arereplacedwithobjectsof32realcomponents.TheK¨ahlerequation—replacementfortheDiracequation—isamostnaturalone(readunavoidable)intheK¨ahlercalculus.3.Itonlyusesscalar-valuednesstotakecareofwhattheDiraccalculusdoeswithvector-valuedness,eventhoughothervaluednessesareallowedandshouldbeused.4.Init,spinisatparwithorbitalangularmomentumandneednotbeforcedintobeingbyinvokingitasaninternalpropertyofparticles.Spin,beingaboutrotations,cannotbeidentifiedwithtorsion,whichisaboutthebreakingofintegrabilityofthetranslationpartoftheconnection.K¨ahlershowsthatspinissimplyaboutknowinghowtoperformpartialderivativesofdifferentialformswhentwosystemsofcoordinatesareinvolved.5.Init,particlesandantiparticlesemergewiththesamesignofenergy.ThisisadirectconsequenceofsplittingintotwopiecesthedifferentialformsthatsatisfytheK¨ahlerequation.6.Foldy-Wouthuysentransformationsareunnecessarytoobtaintheclas-sicalelectromagneticHamiltonianbeyondthePauliterms.Inspiteofwhatisclaimedbyitspractitioners,thosetransformationsdonotseparateparticles

262244CHAPTER13.ABOOKOFFAREWELLSfromantiparticlesbutoddandevenpartsoftheprimordialfieldinwhichanelectronorapositroncontainthedominantamountofenergyinthatfield.7.Thedifferenceof2mc2betweentheactualenergymc2ofapositronandtheenergy−mc2thattheDiractheoryassignstoitintheDiractheoryiseasilyexplainedbythesameprocessasin#3.Oncewehavemadethecaseforit,letusseeitsimplicationsforthefoun-dationsofquantumphysics.A.Thefieldistheprimordialconcept.InaStern-Gerlachexperiment,wearecertainlychangingtheorientationofspin,butthisisaderivedconcept.Whatchangesisthefielditselfofwhichtheelectronsformpart.Itentailsthechangeintheconcentrationsoftheenergyofthefieldthatweidentifyastheelectronsinthebeam.B.Particlesemergefromtheprimordialfieldasitscomponents(inspecialcases)whendecomposedintomembersofideals.Inaprocessinwhichapairiscreated,orwhenparticlescomeouttogetherinadeepinelasticexperiment,theseareentangledabinitiobyvirtueofthefactthattheyemergefromthedecompositionofsomestateoftheprimordialfield.C.Hence,aninternalpropertyfromaparticleperspectiveisstandardprop-ertimepropertyfromafieldperspective,andis,therefore,aspacetimeproperty.D.U(1)andPoincar´esymmetrycometogetherinthesplittingofthealgebraofprimordialfieldsintoideals.Hence,chargeemergesinaspacetimecontext.Inotherwords,chargeemergesfromtheinnerdynamicsofbundlesrelatedtothetangentbundle,notrelatedtoauxiliarybundles.Seeitem(a)below.E.Thesymmetriesthatdefineparticlesarenotsymmetriesoftheequationsoftheprimordialfield,orevenoftheprimordialfielditself.Theyarepropertiesofspinorsolutionsofthoseequations.Inotherwords,theyaresymmetriesofmembersofcertainideals.Hence,weshouldlookatthesymmetriesoftheidempotentsthatdefinethoseideals.Thesymmetriesthatwecanembodysimultaneouslyintoidempotentsmatter,notthesymmetriesoftheequationsthatthoseidempotentssatisfy,whicharealwayslarger.F.TheproperwaytodorotationsofspinorsisthroughtangentCliffordalgebra.Thevaluednessesofthedifferentialformsinthetheoryofthemovingframe—valuednessesofwhichwespokeinsection5ofchapter12—isnowextendedintoaCliffordvaluedness.Connectionshavetotakecareofthat,whichmakesthearenaofacorrespondingtheoryofconnectionsthesameasthearenaforK¨ahler’stheorybeyondscalar-valuedness,andthusbeyondtheelectromagneticinteraction.G.Whentheunitimaginaryisreplacedbyappropriaterealgeometricquan-tities,theothersocalledinternalsymmetriesstarttoemergeinaspacetimerelatedcontext.Itis“spacetimerelatedcontext”andnotsimply“spacetimecontext”,sincethetheoryofthemovingframeonmanifoldsisnotallthatthereistothetheoryofconnections.Particlesandphysicalfieldsarenotrepresentedindepthinthetheoryofmovingframes.Aspace-propertimesubspaceofa5-DKaluza-Kleinspacewithoutcompactificationcomestotheforeforquantummechanics[89].H.ThisKaluza-Kleinspaceexistsonlyinanappropriate“preferredframe

26313.6.FAREWELLTOAUXILIARYBUNDLES?245context”,eventhoughtheLorentztransformationsretaintheirvalueforsignifi-cantspecificpurposes.TheLorentzmetric,nowexpressedasaCliffordalgebraequation,playstheroleofnaturalliftingconditionforcurvesintime-space-propertime.I.Preferredframestogetherwithametriccanonicallydetermineteleparallelconnections,notLevi-Civitaconnections.Bothtypesofconnectionsareequallyconsistentwiththesamemetric,asonlytherelationsofaffinetypechange.Theyfavorteleparallelismsinceitentailsequalityofvectorsatadistance,whichLevi-Civitaconnectionsdonot.Inaddition,teleparallelismbringsnewdegreesoffreedom,embodiedinthetorsion.LetusnowshowhowCliffordalgebraandtheK¨ahler’stheorybasedonittakeusbeyondtheelectromagneticinteractionwhenweletthemathematicsspeak.Asafirststepandbyconsiderationsrelatedtosolutionswithtimetransla-tionandrotationalsymmetries,thefactoreimφ/−iE0t/insolutionswithpropervaluesofenergyandangularmomentumforelectronsandpositronsnowhastobereplacedwithimφ/−iE0t/11e(1±idt)(1∗idxdy),(6.1)22wherethestarmeans±and∓.So,astartingpointtofindsuchpropersolutionsistowritetheseintheformu=eimφ/−iE0t/p∨τ±∨∗,(6.2)where±1±1τ≡(1±idxdy),≡(1∓idt),(6.3)22andwherepisadifferentialformthatdependsonlyon(ρ,z,dρ,dz)[47].Idem-potents(6.3)representthetwopropertiesofparticlesthataredeterminedbyrepresentationsofthePoincar´egroup,massandspin,hereaccessedondifferentgrounds.Keyalsoisthefact,notobviousifonehasnotstatedtheproofof#5,thatchargecomesintrinsicallyunitedwithenergy,sincethetwosignsin±correspondtothetwosignsofcharge[48].Hence,intheelectromagneticsolutionswhichareproperfunctions,U(1)ispresentinthefactor±,butthereistheadditionalfactorτ±inordertomakeparticles.WeshallcutthestoryshortandproposethatU(1)×SU(2)isrepresentedbythe12idempotents±∗±∗±∗ετyz,ετzx,ετxz,(6.4)where±1ij±1τij≡(1±aiajdxdx),ε≡(1∓wdτ),(6.5)22inthesamewayasτ±∨∗representstheelectromagneticinteraction.iandjaretwoconsecutivenumbersorderedasin1,2,3,1,andwistheunitvectordualtopropertime.

264246CHAPTER13.ABOOKOFFAREWELLSOnecannot,however,representparticlesasin(6.2)using(6.5)atthesametime.Thepropervenueforsuchapossibilityrequiresfurtherstepsthatinvolveabinitioquarks.Toobtainthenitsufficestomultiply(6.4)byanotheridem-potentthattakesthreevalues.Butletusstopthere.Wehavesaidenoughformakingthepointthatthereisafutureinbiddingfarewelltoallcalculusexceptafurtherenrichedcalculusofdifferentialforms.Withthese,weapproachtherealizationofEinstein’sthesisoflogicalhomogeneityoftheoreticalphysicsandtangentbundledifferentialgeometryatthesametimeascalculusalsobecomestangentbundlegeometry,andthisgeometrybecomescalculus.Ifthisisnotthegermofafarewelltoauxiliarybundles,whatis?

265AppendixAGEOMETRYOFCURVESANDSURFACESA.1IntroductionCurvesandsurfacesaredifferentiablemanifoldsofdimensionsoneandtworespectively,exceptiftheyself-intersect.Inthiscasewewouldcutthemintopiecesattheintersections,inordertohavemanifolds.Doingsochangestheglobalpropertiesofthesetinquestion,notthelocalones,except,again,attheintersections.TheGauss-Bonnettheoremisanexception,meaningthatitis(orcanbemadeinto)aresultofglobaldifferentialgeometry.Weareinterestedinlocalproperties.Connectionsarelocalconcepts.Amoreimportantdifferenceisthatweshalldealwith1-Dand2-Ddifferen-tiablemanifoldsembeddedin3-DEuclideanspace,asopposedtothosemani-foldsonthemselves.Thinkoftheconcept“embedded”asifitwere“immersed”intheordinarysenseoftheterm(Technically,embeddingisaninjectiveim-mersion).Butforgetaboutthetechnicalmeaningsandlookattheissueinthefollowingsimpleterms.Inthetheoryofcurvesandsurfaces,wedonotthinkofthemaswedowhendealingwithdifferentiablemanifoldsofdimensionsoneandtwo;acurvewouldthenhavezerocurvatureandtorsionsincedifferentialtwoformsarezeroindimensionone.Andyetwespeakofcurvatureandtorsionofcurvesthatarenotzeroingeneral.So,wearedealingwithalternativeconceptsoftorsionandcurvature,sincevectorsthatarenottangent,speciallyandspecificallynormalvectors,arealsoconsidered.Theruletorelatevectorsatdifferentpointsofacurveorasurface—thusthedifferentiationofvectorfields—istheconnectionof3-DEuclideanspace(Thisconnection,beingtrivialintermsofconstantframefields,doesnotmakeitspresencebefelt).Onsurfaces,themetricof3-DEuclideanspacecanbeexpressedintermsofjusttwocoordinates,butatthepriceofmakingthequadraticformmorecomplicatedthantheCartesianone.Computationsonsurfacesarenot,however,247

266248APPENDIXA.GEOMETRYOFCURVESANDSURFACESthattrivial.Ingeneral,thenatureoftheissuesconsideredinvolvesframefieldsadaptedtothesurface,notconstantframefields.Asforthestandardconceptof(Levi-Civita)connectiononasurface,itishereamatterofneglectingthenormalcomponentinthedifferentiationoftangentvectorstothesurface.Ashasbeenthecasethroughthetext,weshallcontinuetousethemov-ingframemethod.Onehasnotyettakenfulladvantageofit.Wedobetterthanappearstobethecaseintheliteraturebytheuseofsurface-adapted3-Dframefieldsthatarecanonicallydeterminedbythediagonalizationofthesecondfundamentalformofsurfaces.WeshallseethattheFrenetframefieldiscanonicallydeterminedbycurvesandthatthe“geodesicframefield”iscanonicallydeterminedbythepairofcurveandsurface.Thereisthecoordinateframefieldonsurfaces,canonicallydeterminedbythepairofsurfaceandcoordinatesystemonit.Therearealsothecanonicalframefieldsof3-DEuclideanspace,i.e.theconstantorthonor-malframefieldsdenotedas(i,j,k).Missingisthecanonicalframefieldofanembeddedsurface.Forcomparisonpurposes,therewillbesomeparagraphswhicharenotproperpartofthetheoryofcurvesandsurfaceswiththetheoryofmovingframes.Theyillustratethecumbersomenessofthebynowalmostabandonedtraditionalapproach.Ifweremovedthosecomparisons,ourtreatmentwouldbemorethantwopagesshorter.A.2Surfacesin3-DEuclideanspaceA.2.1Representationsofsurfaces;metricsOnewayofgivingasurfaceisintheformx=xi(u,v)a,(2.1a)iwhereaiisafixedbasis,i,j,k.Equivalently,wehavex=x(u,v),y=x(u,v),z=z(u,v),(2.1b)foragivendomainoftheparameters.Theseactuallyarecoordinatesonthesurface.Wemayalsogiveitasf(x,y,z)=0.(2.2)Theconnectionbetween(2.1)and(2.2)canbeestablishedasfollows.Wesolveforuandvintwooftheequations(2.1b)andsubstituteinthethirdofthoseequations,thentakingalltermstothelefthandside.Ontheotherhand,wegobackfrom(2.2)to(2.1)bywriting,forinstance,x=u,y=v.(2.3)Substitutionof(2.3)in(2.2)allowsustoviewzasanimplicitfunctionofxandy.Ofcourse,onemaythenuseotherparametrizationsthroughaninvertible

267A.2.SURFACESIN3-DEUCLIDEANSPACE249systemofequationsu=u(u,v),v=v(u,v).(2.4)Themetricof3-DEuclideanspaceintermsofCartesiansystemsisds2=(dxi)2,i=1,2,3.(2.5)iwheredxi=xi,duα,α=1,2(2.6)αandwherexi=(x,y,z)andu1=u,u2=v.Asubscriptafteracommameanspartialdifferentiation.Thevariablewithrespecttowhichthedifferentiationtakesplaceisapparentinanysuchexpressions.Oncetheseoperationshavebeenperformed,anytraceofx,y,z,dx,dyanddzhasdisappearedandweareleftwithds2=g(uγ)duαduβ,(2.7)αβwhichisalsowrittenasds2=Edu2+2Fdudv+Gdv2.(2.8)Itisthecalledthefirstfundamentalform.ClearlyE=g11,F=g12,G=g22.(2.9)ThepositivedefinitenessofthemetricimpliesEG−F2>0,(2.10)asweareabouttoshow.(2.8)isthewayinwhichoneexpressesdistanceoncurves,whichisactuallycomputedas#22dududvdvE+2F+Gdλ,(2.11)dλdλdλdλwhereλisaparameteronthecurve.Ingeneral,wecantaketheparametertobevitself.(2.11)thenbecomesEμ2+2Fμ+Gdv,(2.12)whereμisdu/dv.Theequation2y=Ex+2Fx+G(2.13)isaparabola.Ifthemetricispositivedefinite,thisisanuprightparabolawithvertexabovethexaxis.Itdoesnotcuttheliney=0.ThustherootsofEx2+2Fx+G=0mustbeimaginary,whichimplies2EG−F>0.(2.14)

268250APPENDIXA.GEOMETRYOFCURVESANDSURFACESA.2.2Normaltoasurface,orthonormalframes,areaVectorsnormaltoasurfacearenottangentvectors.Hence,theycannotbegivenasalinearcombinationoftangentvectors,butratherintermsof,forexample,thebasisai=i,j,k.Giventhesurfacef(x,y,z)=0,itstangentplaneat(x,y,z)isobtainedbysubstitutingxi−xifordxiinf,dxi=0,i.e.0000if,·(xi−xi)=0,(2.15)i0wherethedotisusedforproduct,thuspreemptingtakingxi−xiasargument0off,i.Letg(x,y,z)bedefinedas−f(x,y,z).Theequationg(x,y,z)=0representsthesamesurfaceasf(x,y,z)=0.Buttheunitvectorsf,iaig,iai,(f,i)2(g,i)2areoppositeofeachother.Whetherwetakeoneortheotherastheunitnormaltotheplaneisnotsignificantatthispoint.Differentiating(2.1),wegetdx=dxi(u,v)a=xdu+xdv.(2.16)i,u,vx,uandx,varedefinedbytheserelations.Theyconstituteatangentcoordinatevectorbasisfield.Itisnot,therefore,orthonormalingeneral.Theequationoftheplaneatthepointofcoordinates(u0,v0)isgivenbyx=x0+x,u·(u−u0)+x,v·(v−v0),(2.17)wherex0,x,uandx,vareevaluatedat(u0,v0).Thedotsatmidheighthavebeenputforthesamereasonasin(2.15).Inmovingframenotation,theyaretheequivalentofiidP=ωei,P=P0+Xei.(2.18)Thefollowingisanecessarysteptolaunchthemovingframemethodonce12theequationofasurfacehasbeengiven.Let(ˆω,ωˆ)beanypairofdifferential1-formsthatsatisfiesds2=g(u,v)duαduβ=(ˆω1)2+(ˆω2)2.(2.19)αβAfirst(retrospectivelyveryimportant)consequenceofthenormalizedformof3themetricisthatˆωiszeroonthesurface.Hence,thepull-backtothesurfaceofthetorsionofEuclideanspace(torsionwhich,ofcourse,iszero)reads3α33αβ0=dωˆ=ˆω∧ωˆα=Γαβωˆ∧ωˆ,(2.20)3where,forthelaststep,wehaveusedthatthepull-backofˆωαisalinearfunctionofonlyˆω1andˆω2.WeimmediatelygetΓ3=Γ3,whichweprefertowriteas1221Γ1=Γ2(2.21)3231(onthesurface).

269A.2.SURFACESIN3-DEUCLIDEANSPACE251Wenowproceedtodiagonalizetheexpressiongduαduβ≡gdu2+2gdudv+gdv2,(2.22)αβ111222byfirstobtaining1gg−g2gduαduβ−(gdu+gdv)2=dv2221112,(2.23)αβ1112g11g11asperEq.(4.51)ofchapter3.Wethushave22αβg11du+g12dv2g22g11−g12gαβdudu=√+dv,(2.24)g11g11andgdu+gdv(gg−g2)1/2ωˆ1=1112,ωˆ2=dv221112(2.25)√√g11g11and,further,ωˆ1∧ωˆ2=(gg−g2)1/2du∧dv=EG−F2du∧dv.(2.26)22111212Clearly,thereisaninfinitenumberofpairs(ˆω,ωˆ)thatorthogonalizethemetricofthesurface.(2.25)isjustoneofthem.Weshallusethisfreedomtodetermineframefieldsadaptedtosurfaces.A.2.3TheequationsofGaussandWeingartenTheconnectionequationsofEuclideanspacearesimplydai=0intermsofaCartesianframefield.TheequationsofGaussandWeingartenaretheconnec-tionequationsof3-DEuclideanspacewhentheframefieldisconstitutedbyx,u,x,vandtheunitvectorNperpendiculartotheplanethattheydetermine.Atatimewhentheconceptofconnectionequationsdidnotexist,Gausswrotethedifferentiationofthetangentframefield(x,u,x,v)as12x,u,u=Γ11x,u+Γ11x,v+eN,(2.27a)x=Γ1x+Γ2x+fN,(2.27b),u,v12,u12,v12x,v,v=Γ22x,u+Γ22x,v+gN,(2.27c)whichreflectsthestateoftheartofthetime.d(x,u)isadifferential2-formonthesurface,valuedin3-DEuclideanvectorspace.Itcanbewrittenasthefollowingbilinearcombinationd(x)=Γ1duαx+Γ2duαx+Γ3duαN(2.28),u1α,u1α,v1α(andsimilarlyfordx,v).Herewecanreadthat123x,u,u=Γ11x,u+Γ11x,v+Γ11N,(2.29)(andsimilarlyforx,u,vandx,v,v).

270252APPENDIXA.GEOMETRYOFCURVESANDSURFACESThetwoWeingartenequationsarefF−eGeF−fEN,u=EG−F2x,u+EG−F2x,v,(2.30a)gF−fGfF−gEN,v=EG−F2x,u+EG−F2x,v.(2.30b)Intermsoforthonormalframes,wewouldreplace(a)x,uandx,vwithˆeα,(b)(e,f,g)withΓ’sasstatedabove.Weshalllaterseehowtheywereintroducedoriginally.(c)(E,G,F)with(1,1,0).Noticethegreatsimplificationthatthenresults.Furthersimplificationfol-lowsinsection4whenweadoptthecanonicalframefieldofsurfaces.A.3Curvesin3-DEuclideanspaceA.3.1Frenet’sframefieldandformulasCurvesinE3are1-dimensionalmanifolds,ifweeliminatethepointswherethecurveintersectsitself.Theycanbegivenastheintersectionoftwosurfacesf(x,y,z)=0,g(x,y,z)=0,(3.1)orasx=x(u),y=y(u),z=z(u),(3.2)whichisthedevelopedformofx=xi(u)a.(3.3)iTherelationbetween(3.1)and(3.2)isobtainedbysolvingfortwooftheCartesiancoordinatesin(3.1),andthenchangingparameterifonesowishes.Intheoppositedirection,weeliminateuin(3.2)Thedistanceonthecurvebetweentwopointsonthesameisobtainedbyintegrationof#222dxdydzds=++du.(3.4)dududuFrenet’sframeisthe3-Dframefieldadaptedtoacurvethat,ifwerequireittobeorthonormal,iscanonicallydeterminedbythecurve.Thefirstelementoftheframeistheunittangentvectort,dxdx1t≡=,(3.5)dsduds/du

271A.3.CURVESIN3-DEUCLIDEANSPACE253withds/duobtainedfrom(3.4).tisobviouslyaunitvectort·t=1,(3.6)sincedx·dxisds2.Thetangentlinetoacurveatitspointxis0x=x0+ts,(3.7)sbeingassignedthevaluezeroatx0.From(3.6),wedefineacurvaturevectorκ≡dt.Wethenhave:dsdtt·κ=·κ=0(3.8)dswhichmeansthattheequations1κ=κn=n(3.9)ρdefinen,κandρifwedemandthatnbeaunitvector.Thisvectoristhesecondelementofarighthandedorthonormalframe.Onereferstoκandρrespectivelyascurvatureandradiusofcurvature.Becauseofthesoughtorthonormalityoftheframe,thecoefficientofnrelativetobwillbethenegativeofthecoefficientofbrelativeton.Thus:n=−κt+τb,b=−τn.(3.10)τiscalledtorsionofthecurve.Equations(3.9)and(3.10)arecalledtheFrenetequations.bisperpendiculartotheplaneofplanecurves.Itisthenofconstantdirectionandactuallyaconstantvector,sinceitisofunitsize.τis,therefore,zero.Planecurvesaretorsionless.Ontheotherhand,ahelixhastorsionsincetheplaneoftandnkeepschanging.Thisplaneiscalledtheosculatingplane.A.3.2GeodesicframefieldsandformulasWearegoingtostudysurfacesthroughcurvesonthem.Itisthenusefultoconsider3-Drighthandedorthonormalframes(uˆi=t,u,N).uisavectorinthetangentplaneandNisusuallytakenontheoppositeside(ofthetangentplane)toκ(seeexampleinnextparagraph).Weshallrefertothisframefieldasthegeodesicframefieldofthecurveonthegivensurface.Onedefinesthenormalandgeodesiccurvaturesκnandκgofthecurveatapointbymeansofthedecompositiondtt≡=κ≡κn+κg=−κnN+κgu.(3.11)dsWefollowStruikandusethesubscript“n”eventhoughtheprojectionisonN,notonn(see(3.9)).WehaveintroducedtheminussigninfrontofκnNsothat

272254APPENDIXA.GEOMETRYOFCURVESANDSURFACESκnbepositiveunderthemostusualconventionwhenchoosingthedirectionofthenormaltothesphere.Thusthenormalntoaparallelisinwards,whereasthenormalatthesamepointtothesphereitselfisusuallytakentobeoutwards.Thisexampleisalsousefulinconnectionwithhowtheradiiofcurvatureoftheparallelandofthespherearerelated.Recallthatthemagnitudesofthecurvaturesandtheirassociatedradiiaretheinversesofeachother.OrthonormalitydeterminesthetcomponentofN.Wethushave,dNN≡=κnt+τgu,(3.12)dswhichdefinesthecoefficientτg,calledgeodesictorsionofthecurve.Usingagainorthonormality,weobtainduu≡=−κgt−τgN.(3.13)dsA.4Curvesonsurfacesin3-DEuclideanspaceA.4.1CanonicalframefieldofasurfaceWeclearlyhavedtdNdxdNκn=−κ·N=−·N=t·=·.(4.1)dsdsdsdsdx·dNisaquadraticsymmetricdifferentialformonduanddv:22dx·dN=edu+2fdudv+gdv.(4.2)Itiscalledthesecondfundamentalform.Wedonothaveanyuseforthisform,asthereisamuchbetterwaytostudynormalcurvaturethroughtheintroductionoftheconceptofcanonicalframefieldofasurface.Letuˆαdenoteanyorthonormal,counterclockwise2-Dframetangenttothesurfaceandletuˆ3beN.Sinceonlyzeroformsand1-formsliveonacurve,thelastonesaremultiplesofeachother.Formaldivisionbyadifferential1-formisjustifiedandwegetωˆαuˆωˆiuˆΓˆ1(ˆω1)2+(Γˆ1+Γˆ2)ˆω1ωˆ2+Γˆ2(ˆω2)2κ=α·3i=31323132.(4.3)ndsdsds2Equation(4.3)appliestoaone-dimensionalmanifoldofframesrelatedateachpointbyrotationsaroundthenormaltothesurface.Wereservetheuseofunprimedquantitiesfortheorthonormalframe(ˆe1,ˆe2,ˆe3)thatsatisfiesΓˆ1=−Γˆ2,(4.4)3231and,becauseof(4.4),alsoΓˆ1=Γˆ2=0.(4.5)3231

273A.4.CURVESONSURFACESIN3-DEUCLIDEANSPACE255Theframesodefined,(ˆe)isuniqueuptoreversionofaxes,since(4.4)isaniequationappliedtoaone-parameterset.Thiswillbecomeclearinthefollowing,atthesametimeasweindicatehowtoobtainit.Weshallrefertoitasthecanonicalframefieldofthesurface.ThenormalcurvaturethentakestheverysimplifiedformΓˆ1(ˆω1)2+Γˆ2(ˆω2)23132κn=.(4.6)ds2A.4.2Principalandtotalcurvatures;umbilicsInspectionof(4.6)showsthattheextremevaluesofκnareκ=Γˆ1,κ=Γˆ2(4.7)131232whencomputedintheˆeiframefield.Theequationsαωˆ=0,(4.8)giveeachoneofthedirectionsofprincipalcurvature(ds2reducesto(ˆω2)2and(ˆω1)2inthosedirections).Foranyothercurve,thedirectionisdeterminedby12theratioˆω/ωˆforthatcurve.Linesofcurvatureisthetermusedtorefertothosecurveswhichgointhedirectionofprincipalcurvatureatalltheirpoints.Theyareobtainedbyintegratingthedifferentialequations(4.8),oneforeachcurve.Theproductκ1κ2iscalledtheGaussortotalcurvatureKofthesurface.PointswhereΓˆ1equalsΓˆ2donotdetermineavalueforκ,whichthus3132nisthesameinalldirections.Thosepointsarecalledumbilics.Allmaximumcirclesthroughapointonaspherehavethesamenormalcurvature.Allpointsofasphereareumbilical.Onaellipse,therearefourumbilicsifallthreeaxesaredifferent.Thenumberofumbilicsisafunctionofthesurfaceandisusuallyfinite.Readerswhowouldfeelmorecomfortablewithamoreconventionalapproachtoextremevaluesofthenormalcurvatureneedonlyformallydivide(4.3)by(ˆω2)2.Itthentakestheform21aλ+bλ+eωˆκn=2,λ≡2.(4.9)λ+1ωˆEquatingthederivativetozero,weget2aλ=2eλ.Ifaisdifferentfrome,1theratioλequalszero,whichmeansthatwegetthecurveˆω=0.The2parametrizationin(4.9)excludesthecurveˆω=0,whichhappenstoalsoyieldoneofthetwoextremevaluesofκn.Ofcourse,itisobtainedbydefiningλas21ωˆ/ωˆ.Ifaequalse,theequation2aλ=2eλissatisfiedforanyλ,whichmeansthatweareatanumbilicalpoint.

274256APPENDIXA.GEOMETRYOFCURVESANDSURFACESA.4.3Euler’s,Meusnier’sandRodrigues’estheoremsWemaywrite(4.6)as(ˆω1)2(ˆω2)2κ=κ+κ=κcos2θ+κsin2θ.(4.10)n1ds22ds212ThisequationisknownasEuler’stheorem.SincetheinputthateachcurveprovidesinEq.(4.10)isjustitsdirection,acorollaryknownasMeusnier’stheoremimmediatelyfollows.Itreads:allcurvesthroughapointtangenttothesamedirectionhavethesamenormalcurvature.Inthecanonicalframefield,thethirdconnectionequationreadsdN=Γ1eωˆ1+Γ2eωˆ2.(4.11)3113222Considerˆω=0,i.e.,Eq.(4.11)thenbecomes11dN=Γ31e1ωˆ=κ1tds=κ1dx,(4.12)2andsimilarlyfortheˆωlineofcurvaturewithκ2.HencewehaveinbothcasesdN−κdx=0.(4.13)ThisequationisknownasRodriguesequation.ComparisonofdN=κitdswith(3.12)impliesRodrigues’estheorem:thegeodesictorsioniszeroforthelinesofcurvatureandonlythey.A.4.4Levi-Civitaconnectioninducedfrom3-DEuclideanspaceWeshallusethesymbolsDanddfortheexterior(covariant)derivativeundertheconnectionsoftheEuclideanspaceandofasurfaceembeddedinit,respec-tively.Needlesstosay,DDiszerobutddisnot.DinducesonthesurfacetheLevi-Civitaconnection,asweshallnowsee.ForanyorthonormalframefieldofE3,wehaveˆω3=0and3jβ3αDˆeα=ˆωαˆej=ˆωαˆeβ+ˆωαˆe3,Dˆe3=ˆω3ˆeα.(4.14)βBecauseoforthonormalityand,thus,skew-symmetry,theonlyindependentˆωα2differentfromzeroisˆω1.βLetusverifythatˆωαsointroducedisindeedtheLevi-Civitaconnectionofthesurface.ItisaEuclideanconnectionbecause,inadditiontoskew-symmetry,itstorsioniszero.Indeed,inanyframefieldwehaveαiααβα3α0=dωˆ−ωˆ∧ωˆi=dωˆ−ωˆ∧ωˆβ−ωˆ∧ωˆ3(4.15)forthefirsttwotorsioncomponentsinEuclideanspace.Sincethelastterm3of(4.15)iszero(becausetheˆωofthesurfaceiszero),weobtainthedesiredresult.

275A.4.CURVESONSURFACESIN3-DEUCLIDEANSPACE257A.4.5TheoremaegregiumandCodazziequationsWenowcomputeDDˆeβmoduloˆe3.Weeasilygetijiαγα3α0=DDˆeβ=(dωˆβ−ωˆβ∧ωˆj)ˆei=(dωˆβ−ωˆβ∧ωˆγ−ωˆβ∧ωˆ3)ˆeα,modˆe3.(4.16)Thenullcontentsoftheparenthesis,specializedforβ=1,allowsustowritetheLevi-CivitacurvatureofthesurfaceasΩ2=dωˆα−ωˆγ∧ωˆα=ˆω3∧ωˆ2=−ωˆ1∧ωˆ2=1ββγ1333=−Γˆ1Γˆ2ωˆα∧ωˆβ=−(Γˆ1Γˆ2+Γˆ1Γˆ2)ˆω1∧ωˆ2.(4.17)3α3β31323231Inthecanonicalframefieldofthesurface,wehaveΓˆ1=Γˆ2=0.Hence323121212Ω1=−κ1κ2ωˆ∧ωˆ=−Kωˆ∧ωˆ.(4.18)SincetheLevi-Civitaconnectionofthesurfaceonlydependsonitsmetric,thisequationcontainsthestatementthatthetotalcurvatureisabendinginvariant.Inotherwords,itdoesnotchangeunderdeformationsofthesurfacethatdonotstretchit.ThisstatementisknownasGauss’esegregiumtheorema(excellenttheorem).ItalsotellsusthattheGausscurvaturecoincideswiththeonlyindependentcomponentofΩ2intermsofˆω2∧ωˆ1(thesignisamaterofwhat1unitdifferential2-formonechooses).ThusΩ2isindependentofrotationsinthe1tangentplane.TheCodazziequations∂e∂f1212−=eΓ12+f(Γ12−Γ11)−gΓ11,(4.19a)∂v∂u∂f∂e1212−=eΓ22+f(Γ22−Γ12)−gΓ12(4.19b)∂v∂uareacumbersomewayofwritingthecontentsoftheremainingcurvatureequa-tions.Sincee,fandgareΓ’sforcoordinateframefields(recallEqs.(4.2)andthatdNisde3),onerecognizesin(4.19)thepatternofcurvatureequations.Itisnotworthderivingthemsinceweunderstandwhattheyare.Interestedreadersshouldjustmakefequaltozeroandmakeappropriatetranslationsofsymbolseandg.A.4.6TheGauss-BonnetformulaWedevelop(4.18)furtherbyintegratingonanysimplyconnectedsurface:12222−Kωˆ∧ωˆ=Ω1=dωˆ1=ωˆ1.(4.20)RememberthatΩ2isthecurvatureofthesurface,notoneofthecomponents1curvatureof3-DEuclideanspace.

276258APPENDIXA.GEOMETRYOFCURVESANDSURFACESRecallthe3-Dgeodesicframefield(uˆi=t,u,N).Byitself,(t,u)isa2-Dorthonormalframedefinedbythepairofasurfaceandacurveonit.Itis122notdualtothecanonicalbasisofsolderingforms(ˆω,ˆω).Letˆω1betheLC12connectiononthesurfacerelatedtothesameframefieldas(ˆω,ˆω).Ateachpointofacurve,thedifferentialofucanbeviewedascomposedof2thehorizontalpartgivenbyˆω1,andaverticalpart,dφ,whereφistheanglethatrelatesthecanonicalframefieldofthesurfacetothegeodesicframefield12ofaparticularcurve.Similarlyforˆω2(=ˆω1),whichismorecloselyrelatedtodu.Inspectionof(3.13)allowsustowrite2κgds=ˆω1+dφ,(4.21)and12−Kωˆ∧ωˆ=κgds−dφ.(4.22)So,finally,12κgds=−Kωˆ∧ωˆ+2π,(4.23)whichistheGauss-Bonnetformula."Inthecaseoftheplane,Kbeingzero,κgdsis2πonanyclosedcurve,provideditisjustaoneloopcurve.Itmightbe2nπforannloopclosedcurve.Oritmightbezeroifaftergoingaroundoneloopandjustbeforeclosing,itreturnsonalmostthesamepathbeforeclosingwhereitstarted.Result(4.23)islocalinprinciple,inthesensethatitisnotaglobalresult.Butaglobalresultcanbeobtainedifone“cuts”asurfaceinsuchawaythatitremainsconnectedandallofitisinsideaclosedcurve,andthenappliesthetheorem.Ontherighthandsideof(4.23),wemayhave,insteadof2π,thevalue0(sphere)or2π(torus),etc.Itdependsonwhatisthecurveneededtoenclosethewholesurface.Fordetailedapplicationsofthistheorem,werefertheuninitiatedinclassicaldifferentialgeometrytothebeautifulbookbyStruik[70].Thisglobalresultcaninturnbeconvertedintoatopologicalresultbyac-ceptabledeformationofthesurfacewhichdoesnotchangethelefthandsideof(4.23).Thisdoesnotchangethe2nπontherighteither.Hence−K12ωˆ∧ωˆisatopologicalinvariant,whichcertainlyisanamazingresult,“easily”obtainedwithdifferentialforms.Foranotherapplicationconsiderpolygons.Theirsidesmaybecurvilinear;infacttheywillifthesurfaceisnotplane.For(4.23)toapply,wewouldhavetosmooththeverticessothatthereisnodiscontinuityinthetangenttothecurve,i.e.sothatithasderivativesandthusaconceptofcurvature.Suchasmoothingwouldbeliketheexitrampofahighspeedhighway,buttightlyplacedtotheactualvertex.Theintegraldφonanysuchasmoothingrepresentstheangleofthechangeofdirectionatthevertex,countingcounterclockwise.Thus,forthesmoothedpolygon,wehaveκgds=κgds+φi,i=1,...,n(4.24)smthplgonpolygoni

277A.4.CURVESONSURFACESIN3-DEUCLIDEANSPACE259wherenisthenumberofsidesofthepolygon.Wewouldthusreplacethelefthandsideof(4.23)withtherighthandsideof(4.24),andthenmoveiφitotherighthandsideoftheequation.Inthisway,weobtain12κgds+Kωˆ∧ωˆ=2π−φi,(4.25)polygoniwhichistheforminwhich,becauseofconnectionwithancestralgeometryandtopology,thetheoremisstated.Letusspeakofonesuchsimpleconnection.Thosechangesofdirectionareexteriorangles,formedbytheprolongationofasidewiththenext.Theyarerelatedtotheinteriorangles,θi,byφi=π−θi.Considerastandardtriangle(rectilinearsides)intheplane.Thelefthandsideof(4.25)becauseκgdoesnotchangeinthestraightpartsandKiszerointheplane.Therighthandsideyields30=2π−(π−θi).(4.26)i=1Wethusgetthewellknownresultthatthesumofthethreeinterioranglesisπ.A.4.7Computationofthe“extrinsicconnection”ofasurfaceGivenasurfacein3-DEuclideanspace,Cartan’smethodforobtainingbyin-spectionthe3-Dconnectionadaptedtoit(orbyanyothermethodforthatmatter)isnotapplicable.Thereasonisthatthesurfacebyitselfdoesnotde-termineaframefieldinEuclideanspace.Forexample,aplaneisacoordinatesurface(x3constant)foraninfinitenumberofcoordinatesystems,thusforaninfinityofcoordinateframefields,amongthemtheCartesian,cylindricalandsphericalones.Givenasurface,thebasisvectorsx,uandx,varewelldefinedintermsof(i,j,k).Innormalizingthemetric,weobtainanewframefieldbymeansofdux+dvx=ˆωαˆe(A.4.2),u,vαwheretheˆωαaretheonesgivenasˆωαin(2.19).Thisallowsustoobtainˆeαandˆe(thelatterbyvectorproduct),andtheirdifferentials,allintermsofx,3,ux,x×xduanddv.TheΓˆcanbereadonlyafterduanddvarereplaced,v,u,vbytheirexpressionsintermsoftheωα.Wedonotneedtoreadthembutjustdothatreplacementforthenextstep.Weshallthushavedˆe=ωαˆe,theˆω333α3beingzerofororthonormalframefields.Wefinallycometothecanonicalframefield,ˆe.Clearlyˆe=ˆe.Hencei33dˆe=dˆe=ˆωαˆe.Wereplacetheˆeintermsoftheˆe.Thecoefficientsof333ααβdˆeintermsoftheˆearethesoughtˆωα.But,inordertoreadtheΓˆα,wehave3β33βαβαtoexpresstheˆωpresentinˆω3withtheirexpressionsintermsoftheˆω.

278260APPENDIXA.GEOMETRYOFCURVESANDSURFACESRecallthattheprincipalcurvaturesareΓˆ1andΓˆ2.Thereisnogreat3132advantageinthiscomputationoveramoretraditionalone.Butthelogicandstructuralsimplicityallowsonetomorereadilyrecollecttheprocesstoderiveanythingofrelevance,andtheprincipalcurvaturesinparticular.

279AppendixB“BIOGRAPHIES”(“PUBLI”GRAPHIES)B.1ElieJosephCartan(1869–1951)B.1.1IntroductionSincethisreportisnotabiographyproper,wereferreaderstoonebyJ.J.O’ConnorandE.F.Robertson[56]ascomplement.Inthisappendix,weareconcernedonlywithprovidingaperspectiveofCartan’swork.Thisauthorbelievesthatinmattersofamathematic-scientificnature,Car-tanisthegreatesthumanmindtoeverwalkthisearth,wayaboveanybodyfromrecentcenturies(Poincar´e,Einstein,Gauss,Euler,Newton,etc.).Wespecified“recent”inordertomakethecomparisonmeaningful.Weshallmakethecaseforsuchassertionsinpartbycombiningauthoritativeopinionsondifferentar-eas;nobodyisinapositiontoemitthemonalltheareasinwhichCartaninvolvedhimself.AndinpartbyshowingthatCartansprintedduringallhisproductivelife,whichextendeduntilafewyearsbeforehediedof,reportedly,alongillness.Weshallhaveprovedallofthisbytheendofthisreportofhiswork.Attheveryleast,IshallhaveunearthedforyoumuchoftheworkbyCartanaboutwhichyouprobablywereunaware.TheversionownedbythisauthorofhisCompleteWorkslists184publi-cations,atleasttenofwhicharebonafidebooks.Weknowofafewotherpublications.Forinstance,apaperwithhissonHenriisnotcountedamongthem.Asweshallsee,someofCartan’sgreatestcontributionswereburiedasin-troductionstotechnicalpapersonsubjectsoflesserfame.Forcomparisonpurposes,recallthatRicciandLevi-Civitawroteapaperonthetensorcalculusin1901[60].In1899Cartanpublishedhiscreationofthefarsuperiorexte-riorcalculus(recognizedassuchonlyinrecentdecades)asanintroductiontohispaper“AboutsomedifferentialexpressionsandthePfaffproblem”[4].A261

280262APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)mathematicianwouldhavegothisnameinthehistoryofmathematicsjustforcreatingthiscalculus,ItisnowthebasisofdifferentialgeometryandofthetheoriesofLiegroupsandexteriordifferentialsystem.Throughitsuse,Cartanalsoshowedthatthestudyofaspaceattheinfinitesimallevelcangiveinfor-mationaboutitsbehaviorinthelarge.Inotherwords,hecreatedatoolnotonlyfordifferentialgeometry,differentialequationsandLiegroups,butalsoforglobalgeometryandtopology(seeGauss-Bonnetinpreviousappendix).Letusproceedsystematically.OurstartingpointwillbetheclassificationthathasbeenmadeofmostofhispapersintothreelargeareasinhisCompleteWorks[24].Theyarealgebra,differentialequationsanddifferentialgeometry.Otherpapersreproducedtherecanbeclassifiedwithinthefollowingareas,whereCartanclaimedtoalsohavepublished:complexnumbers,topology,integralinvariantsandmechanics,andrelativity.Seehis“Noticesurlestravauxscientifiques”,whichhewrotein1931,tobefoundasthefirstpaperin[24].B.1.2AlgebraInalgebra,heclassifiedLiegroupsandLiealgebras,bothrealandcomplex,simpleandsemisimple.Hegavetheirrepresentations.Heburiedhisdiscoveryofspinors—likehedidsooftenwithotherimportantdiscoveries—ina1913paperonprojectivegroups,whichincludedthestudyofthelinearrepresentationsofsimplegroups[7].ItwasonlyafterDiracshowedtheirrelevanceinphysicsthatspinorsreceivedexplicitattentioninCartan’swork[23].CartanwasamajorcontributortothefieldofCliffordalgebras,notonlybecauseofspinorsbutalsobecauseofthesocalledmultiplicityof8andtriality.Butthereismuchmore,havingtodowithmethodandlanguage,whichfacilitatethemakingoffurtherdiscoveries.LetushearfromthelatephilosopherandmathematicianRotaandcollaboratorsBarnabeiandBrini[1].Writingin1985aboutGrassmann,whowasanothersupergenius,theysaidthis:“Thereisstrongevidencetoindicatethatmostmainalgebraistsofthetime,suchasGordan,Capelli,Hermite,Cayley,SilvesterandevenHilbert...didnotrealizethesweepingextentofGrassmann’sdiscoveryanditsrelevanceininvarianttheory.Theepigonsofinvarianttheoryinthiscentury,suchmathematiciansasTurnbull,Aitken,AlfredYoung,LittlewoodandevenHermannWeyl,perpetuatedthesamesinofomission,andonefindsintheseauthors’workscatteredrediscoveriesandpartialglimpsesofideasthatcouldhavemadetobloom,hadtheauthorsusedevenonlythenotationofexterioralgebra.”(Yes,DavidHilbertandHermannWeylmissedexterioralgebra).Rotaetal.thenproceedtogivecreditto:Clifford,Schr¨oder,Whitehead,CartanandPeano.TheythencreditCartanspecificallywith:

281B.1.ELIEJOSEPHCARTAN(1869–1951)263“Herealizedtheusefulnessofthenotionofexterioralgebrainhistheoryofintegralinvariants,whichwaslatertoturnintothepotenttheoryof‘differentialforms.”Weshallspeakofhisworkonintegralinvariantsfurtherbelow.But,inviewofwhathasjustbeensaid,doeshenotqualifyasthegreatestalgebraistofhistime,likeGrassmannwasbeforehim?B.1.3ExteriordifferentialsystemsAnysystemofordinaryorpartialdifferentialequationscanbewrittenasanexteriordifferentialsystem.Aroundtheturnofthecentury,IaskedanotherparticipantinameetingondifferentialgeometryoftheSouthEastoftheUSAthefollowingquestion.Howisthefieldofpartialdifferentialequationsstruc-tured?Herespondedsomethinglikethis“Theworkinthisfieldiscomprisedoftwoparts.OnepartistheCartan-K¨ahlertheoryofdifferentialsystems[45].Theotherpartismadeofeverythingelsethathasbeenwrittenonthesubject.”Intheeighties,someofthetopexpertsinthatfield(Bryant,Chern,Gardner,GoldschmidtandGriffiths[3])joinedforcestoworkthroughandappreciatethisbranchofCartan’swork,theresultofwhichwastheirwork“Exteriordifferentialsystems”.Oneofthem,thelateprofessorGardner,wrotealsoamonographonCartan’smethodofequivalence,methodwhichbelongstothisareaofmathematics.Intheintroduction,hehadthistosay[42]:“ElieCartan’sapproachwasmotivatedbyhisworkoninfinitepseu-dogroups....In1908,inhastobeoneofthemostremarkablepapersinmathematics,‘Thesubgroupsofthecontinuousgroupsoftransforma-tions’,ElieCartanformulatedanddesribedaprocedure...Hedidthisinjust25pagesoutofa137paper...”Those25pagesareapaperwithinapaper.B.1.4Geniusevenifweignorehisworkingonalgebra,exteriorsystemsproperanddifferentialgeometryWenowaddtothemotherexamplesofgenialworkfromhisbookonintegralinvariants[9].Justthatbook,ifitwereexplainedandused,wouldbyitselfmakeCartanoneofthegreatestmathematiciansofalltimeandoneoftheforemostmathematicalphysicistofhistime.Thepaceisdizzying.Chapter1inthatbookstartsbyobtainingtheequa-tionsofmotionfromthevariationalprinciple.Bypage4hehasalreadyobtainedtheHilbertintegralandbypage7hehasderivedtheequationsofmotionfromtheHilbertintegral.Bytheendofthechapter,page16,hehasderivedJacobi’stheoremandhasstarteddiscussionofthecanonicalformoftheequationsofmotioninthepresenceofperturbationforces.Foranothergreatexampleofthecaliberofthisbook,letusjumptoitschapter9,page81.Infourpages,hestatestheconceptofLieoperatoractingonfunctions(whichhecallsinfinitesimaltransformation),heextendsittothe

282264APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)ringofdifferentialforms,explainsitsimpactforobtainingfirstintegrals,givestwowaysinwhichthatoperatorgeneratesnewdifferentialinvariantsfromagivenone,relatesthosetwoandproducesintheprocessformulastogeneratethoseinvariants.Inthesamebook,Cartanprovidesawealthofapplicationstofluids,optics,rigidbodydynamics,manybodyproblems,standardissuesinthefoundationsofclassicalmechanics(likegeneralizationsofthetheoremofPoisson-JacobiandofthePoisson-Jacobibrackets).Fluidsaretreatedlikepartofclassicalmechanics,i.e.followingtheparticlesintheirmotionratherthanintermsofwhathappensatanygivenpoint.Resultsemergeinquicksuccession.InthesamebookandasanapplicationofLieoperatorstothen-bodyproblem,Cartandealswithstandardpotentialenergywheretheexponentoftheradiusinthedenominatorisleftundetermined.Infourpages,whicharebarelyenoughtowritedownhisresults,heobtains11differentialinvariants(whoseintegralsareintegralinvariants,oneofthemresultingintheJacobiintegral).Fromthemheobtainsawealthoffirstintegralsandevenobtainsthetheoremthatstatestheinvarianceoftheproductofthesquareoftheangularmomentumandthetotalenergy,bothinthemovementaroundthecenterofmass.Inonepageheobtainsfromfirstprinciplestheinvarianceofthemagnitudeofthevelocityofthepointsinamovingrigidbody.AfteronegraduatesfromthestudyofthebookonMechanicsbyLandauandLifshitz[51],youmaybesaidtobepreparedforanewlevelofcomprehensionofthesubjectwiththatbookbyCartan.B.1.5DifferentialgeometryGardnercontinuedasfollowsthecontentsofthecitationwehavemadeofhiminsubsection1.3[42].“Theimpactofthese25pagesonKlein’sprogramwaseloquentlyde-scribedbyJ.Dieudonne’whenspeakingofCartan:‘Finally,itisfittingtomentionthemostunexpectedextensionofKlein’sideasindifferentialgeometry...’ConsiderthestatementbyHermannWeylin1938,whichwereproducefromthesameGardnerreference:“‘Cartanisundoubtedlythegreatestlivingmasterindifferentialgeometry.”Why?Cartanreinterpreted,unifiedandgeneralizedtheprogramsofKleinandRiemanningeometry.Usinghisdiscoveriesongroupandalgebrasandontheintegrabilityofdifferentialsystems,hecreatedthetheoriesofaffine,metric,Euclidean,projective,conformalconnections,etc.Heindeedshowedthatvirtuallyallothereffortsingeometryfitintohisscheme.Intheprocess,

283B.1.ELIEJOSEPHCARTAN(1869–1951)265hegavethedifferentialformofEuclid’spostulatesinspacesofanynumberofdimensions.CartanalsoauthoredthemostimportantpaperinFinslergeometry,whichgaverisetotheconnectionnowknownastheCartan-Finslerconnection.Hecreatedtheconceptoffiberspaces.Infactheworkedwithsuchspaces(equiva-lentlybundles)sinceveryearlyinhisprofessionallife,andittookhalfacenturyfortheconcepttobeunderstood.CreditforbringingsuchanunderstandingisrightlygiventoEhresmannasfarasmathematiciansareconcerned[36].Butphysicistsknowverywellwhattheycallthesetofinertialframes,asithasawellknownandunderstoodbundlestructure.Theyonlyneedtolearnthetechnicalnames.Themethodofthemovingframecarrieshisnamebecauseheiscreditedwithhavingcreatedit.Thisisincorrect.SeveralFrenchmathematicianshadalreadydevelopedandusedit,asCartanhimselfreported.ButhisextremevirtuosityonthissubjectallowedhimtocreateframesappropriateforarbitraryLiegroups,andthenstudythemwiththatmethod.ManyphysicistswhooftenclaimacquaintancewithhisworkhavenotevenunderstoodhisviewofhowRiemanniangeometryistobeseen,namelyaspertainingtofiniteLiegroupsratherthantheinfiniteLiegroupofcoordinatetransformations.Thisiswhytheyfailtoseehowgravitation(whichtheyin-correctlyassociatewiththeinfinitegroupofdiffeomorphisms)couldbeunifiedwiththeotherinteractionswithoutresorttofarfetchedideas.Anotherexampleofmuchformalismandlittlesubstanceinmuchofthemodernliterature:eventechnicalbooksfailtogiveniceexamplesofspaceswithtorsion,inspiteofthefactthatthereareverysimpleexamples(oneofthemgivenbyCartanhimselfin1924,namelywhatwehavecalledtheColumbusconnection).B.1.6CartanthephysicistCartanpublishedmuchonphysicalissues.Wehavealreadyspokenofhisworkinclassicalmechanics.Healsoworkedinotherareasof(classical)physics,workthathasbeentotallyignored.Itmayhavebeenduetoitssophistication(itwaswrittenintermsofdifferentialforms),tobeinginFrenchandtohavingtodowithfoundationalproblemsthathisworkonphysicshasbeentotallyignored.Ifithadbeenstudiedandunderstood,hewouldhavebeenconsideredagreatphysicist.Takeelectrodynamics.Justhisdiscussionofelectrodynamicsinafewpagesofhis1924paper(secondintheseries)onthetheoryofaffineconnectionssuper-sedessimilarworkinapaperof1934byVanDantzigpresentedbyDiractoandpublishedinProceedingsoftheCambridgePhilosophicalSociety[71].InthosefewpagesCartandiscusseswhatistherightwayofwritingMaxwell’sequa-tions,whethertheseequationsdependonconnection,andwhetherelectrody-namicsitself(therearetheenergyequationsinadditiontoMaxwell’s)dependsonconnection.Heevengivestheformoftheelectromagneticenergy-momentumtensorinakindofKaluza-Kleinspacethatheintroducesinaninformalway.Infact,in1923-1924hewroteallthefundamentalphysicsofthetime(exceptthe

284266APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)emergingquantummechanics)intermsofdifferentialforms,withimplicationsthathedidnotmentionastotheemptinessofgeneralcovarianceasageneralprinciple,sinceitisjustaheuristichelp.Considerthefollowing.Mostrelativistsstilldonotunderstandtheconser-vationlawofvector-valuedforms,althoughhegaveitinhis1922onEinstein’sequations.Thisisofparticularimportanceincosmology.WiththeLevi-Civitaconnection,itdoesnotmakesensetointegratethoseformsbecausethatconnec-tiondoesnotallowforpath-independenceidentificationofthetangentspacesatdifferentpoint.Ifonestillobtainsgoodresultsitisbecauseametrictogetherwithapreferredframedeterminesateleparallelconnection.Cosmologists,with-outknowingit,areusingtheteleparallelconnectiondeterminedbythemetrictheyuseandtheframeofreferencecomovingwithmatteratthelargestscales.Butthenspacehastorsion,whichpeoplewithaverypoorknowledgeofgeom-etryandofLiederivativessayitisnotpresentinspacetime.Iftheyjustknewwheretolook.Cartan’spotentiallygreatestcontributiontophysicsmaystillberealizedifandwhentheCartan-Einsteinunificationweretobecomeoneday(partof)theunifiedtheorythateverybodywishesfor.Indeed,CartanalreadyexplainedtoEinstein(butthelatterdidnotunderstand)thedifferencebetweenformulatingageometricunifiedtheoryasaphysicistandasademiurge,specificallyinteleparallelism.Tobeademiurgemeansthatonedemandsofthefieldequationsthattheyprovethattheuniverseiswhatonepostulatesittobe,andnotjustthattheuniversebeconsistentwiththepostulate.Inthecaseofteleparallelism(TP),itmeansthatthefieldequationsmustimplyTP,andnotsimplyignoretheaffinecurvaturebecauseitiszero,whichiswhatEinsteinwasdoing.CartandidinformonthistoEinstein(andactuallyshowedtohimacoupleofequationsinthisregard!,butthelatterdidnotlisten).Theironyisthatiftheequationsofstructureofsomespacearemadefieldequationsofthephysics,thatisthepurestformofimplementationofEinstein’sthesisoflogicalhomogeneityofgeometryandtheoreticalphysics.B.1.7CartanascriticandmathematicaltechnicianCartandidinpassingandmoreefficientlyworkforwhichotherauthorshadreceivedcredit,likeSlebodzinskionLiedifferentiation[68]andWeitzenbock[90]and[91]onteleparallelism.Whitneyisfamousforhisworkonembedding[92].Althoughthesubjectcoveredisfarfrombeingthesame,Cartan’scomparableworkonembeddingisjustonemoreofhispapersandnotevenmentionedintheliterature.Hehadthevisionofmathematicsthatallowedhimtoincontestablysaywhatwaswrongwithotherpeople’sprogramsregardlessoftheirstature,asweretro-spectivelyknow.Thus,forexample,hehadtheperspectiveandauthoritytousetheterm“thefalsespacesofRiemann”(heproceededimmediatelytospeakoftheirredemptionbyLevi-Civita).Inlessdramaticterms,heexpressedpointedcriticismsand/orlimitationsoftheworkofothermathematicians,commentswhichhistoryhasprovedtobecorrect.Suchisthecasewithhiscommentson

285B.1.ELIEJOSEPHCARTAN(1869–1951)267theworkofWeitzenbock,RicciandaFinslergeometerwhosenameweprefernottomention.Seealsonumerousexamplesin“Noticesurlestravauxscien-tifiques”[24],wherehecorrectsevenPoincar´e(page93ofthereport,page107intheCompleteWorks).CartantookapaperbyStudyonsystemsofcomplexnumbers(Altereund¨neuereUntersuchungenuberSystemecomplexerZahlen)andmultiplieditbyfourinsize[6].Thispaperisatremendouspieceofworkonalgebraicdevel-opmentsinthenineteenthcentury.IfyouwanttounderstandthescopeofGrassmann’swork,readthatCartanpaper.Cartanhadthegenialabilityofthinkingofsomewaytoavoidbruteforcemethodsandcomputeinafewlineswhatothersthatforwhichotherswouldtakepages.AfewexamplesarehisderivationoftheequationsofstructureofEuclideanspaceinafewlinesasasimpleexerciseofchangeofcoordinates,hiscompu-tationoftheLiederivativeofdifferentialformsoroftheprimitivesofcloseddifferentials,etc.,etc.[9]JustalittlesteptakenbyhiminhisstudyofLiegroups[22]suggestedourdiagonalizationofthesecondfundamentalformofthetheoryofsurfacesinthepreviousappendix,whichresultsinatremendoussimplificationofthattheory,aswehaveshowninthepreviousappendix.Allthisspeaksof(touseDieudonn´e’swords[33])“hisuncannyalgebraicandgeometricinsightthathasbaffledtwogenerationsofmathematicians”.B.1.8CartanaswriterForsomereason,Cartan’swritingsareviewedasdifficulttounderstand.Iguessthatmuchofithastodowiththeextremelyformalwayinwhichmodernmath-ematiciansaretrained,wherethetrees(definitions,theorems,proofs,lemmas,corollaries,remarks,etc.)andsymbolismimpedeavisionoftheforest(theleadingstory).SometestimonialsaboutthedifficultyinunderstandingCartan,byGardnerabouthimselfandothers:“AfterthinkingaboutthismethodforanothertwentyyearsandtalkingperiodicallywithS.S.Chernandmystudents,especiallyRobertBryant,Irealized...Onemaywonderwhyanyonewouldneedtwentyyearstounderstandapaper.However,IwasnotaloneinexperiencingdifficultywithpartsofthetheoryofRepereMobileandthemethodofequivalence,asthefollowingtwocitationstestify.”ThefirstofthosetwocitationsisduetoHermannWeylinhisreviewofaCartanbook[22].Hesaid:“NeverthelessImustadmitthatIfoundthebook,likemostofCartan’spapers,hardreading...”

286268APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)WeylgoesontowonderwhetherthisdifficultyhastodowithnothavingbeentrainedintheFrenchsystem.ThesecondoccursinthemiddleofSingerandSternberg’spaper[1965]:“WenowresumesomeoftheprincipalformulaeincoordinatenotationwiththeideaofprovidingapartialguidetosomeofthewritingsofE.Cartanontheinfinitegroupsandontheequivalenceproblem.Wemustconfessthatwefindmostofthesepapersextremelyroughgoingandwecannotfollowalltheargumentsindetail.”Cartanhadtheabilitytocomewithsome“trick”tocomputeinafewlineswhatwouldtakeothersseveralpagestodo.Gardner(againintheintroductiontohisbookonthemethodofequivalence)makesapointsimilartotheonebyDieudonn´ementionedintheprevioussubsectionaboutCartanunparalleledcomputationalabilities:“...workingmostoftheexamplesworkedoutbyCartanmadeyoufeelthatspecialtricksandbrilliantobservationswerepartofthemethod.”ThestyleofCartan,likeK¨ahler’s,isthestyleofphysicists.Hedevelopsideasasifheweretellingastory.Hisstyleshouldnotbedifficultforphysicistsifgivensomeguidancebythemathematicianswhonowclaimtounderstandhim.Forthat,mathematiciansshouldexplaintophysicistswhatisgoingonratherthantranslateCartan’sstoriesintoacollectionofdefinitions,theorems,remarks,proofs,corollaries,lemmas,etc.,whichwephysicistssomuchdislike.B.1.9SummaryDifferentialgeometry,topologyandanalysisarenowconnectedthroughthecommonlanguageofdifferentialforms,whichCartancreated.Hewasuniqueinthetheoryofexteriordifferentialsystems,becausehewassogreatinalgebra.Inturnhisawesomecontributionstoclassicaldifferentialgeometryaremadepossiblebyhisprofoundknowledgeofthetheoryofexteriordifferentialsystems.AnditwasontheshouldersofCartanthatK¨ahlerbuilthisworkonthetheoryofpartialdifferentialequationsandhisexterior-interiorcalculus.Cartan’salmostcompleteworkscomprisesmorethan46hundredpages.From1893to1949,bothinclusive,Cartanhadatleastonepublicationeachyear,exceptfor1900(buthepublishedthreein1901),1903(buthehadpublishedthreein1902),1906(buthepublishedtwoin1907),1921(buthehadfourpublicationsin1920andninein1922),and1948(butrememberhediedin1951,ofalongillness,attheageof82).Cartanthescientist/mathematiciansprintedalmosttotheendofthesci-entificmarathonherun,whenforcemajeure(sickness)mayhavecausedhisabandoningtheraceataboutkilometer40.Hewasnotfromthisplanet.Anyclassificationofgeniusesinthemathematics/physicsfieldonlyshowstheig-noranceofthosedoingtheclassificationiftheydonotputhimatleastin

287B.2.HERMANNGRASSMANN(1808–1877)269competitionforthefirstplace.Noothermathematiciansprintedallhislife.Eulerwasanothermarathoner,buthissprintingwasnotquitethesame.Thetopalgebraist,thetopexpertondifferentialequationsandthetopdifferentialgeometer,anextraordinarymathematicaltechnicianandaphysicistwhoseworkonthissubjectmayonedaybeappreciatedwhenunderstood.ThatsummarizeswhoCartanwas.Reportedly,healsowasagreathumanbeingandfamilyman.Thatisthecherryinthecocktail.B.2HermannGrassmann(1808–1877)B.2.1MinibiographyInhisdayjob,HermannGrassmannwasahighschoolteacher.Inhissparetime,hebecameoneofthegreatestmathematiciansofalltime.Inaddition,hewasasuperblinguistwhotranslatedtheHinduRigVeda(literatureandreligion)fromSanskritintoGermanverse,andproducedanextensivedictionaryintheprocess.Hewasanaccomplishedfolklorist,musicianandnaturalscientist(botany,crystallography,colormixing,statics,electricity,acoustics,theoryoftides,inventedaheliostat,etc.).Allthatisstatedinthebook“ANewBranchofMathematics”[43],whichisthemaintranslationofmuchofhisworkwithout,however,constitutinghiscompleteworks.GrassmannundertooksixsemestersofuniversitystudiesinBerlin,hissub-jectbeingphilology,theology,philosophyandpsychology.Reportedly,hedidnottakeanymathematicscoursesthere,butreadbooksinmathematicswrittenbyhisfather,JustusG.Grassmann,anotherhighschoolteacher.Infact,theideaofpurelygeometricproducts,whicharethebasisofH.Grassmann’smathe-maticalwork,washisfather’s.HermanndevelopedJustus’esbasicconceptsintoafullblownsystem.Hepassedseveralstateexaminationswhichqualifiedhimtoteachavarietyofsubjectsinhighschool.HewaseventuallygrantedaDoctorofPhilosophyHonorisCausabytheUniversityofT¨ubingen.HewaselectedacorrespondingmemberoftheG¨ottingenscientificsociety.Afterhisdeath,hewasfinallyrecognizedbyFelixKleinandSophusLie,whowasinstrumentalinthepublicationofhisCompleteWorks.Inmathematics,heiscreditedwithhavingdiscoveredexterioralgebra,buthealsovirtuallydiscoveredCliffordalgebraandadvancedthestudyofprojectivealgebraicvarietieswhileworkingasahighschoolteacher.Andthereismuchmore,asweshallnowintimate.B.2.2MultiplicationsgaloreGrassmanndidnotusethevectorproduct,whichisacontraptionpeculiartothreedimensions,alsotosevendimensionsinamoreartificialway.Aswehaveseeninchapter3,thevectorproductindimensionthreecomprisestheexteriorproduct,butincombinationwithHodgeduality.Itdidnotexistatthetime,but,haditexisted,Grassmannwouldhavevieweditasjustafootnotetohis

288270APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)system,ascanbeinferredfromhiscommentsonHamilton’squaternions.Asinvectoralgebra,onlygradeszeroandonearepresent.Hisextensiveuseofassociative,distributive,commutativeandanticommu-tativelaws(sometimesinmuddledways)wastotallyuncharacteristicinhistime.By1844,hewasdealingwithmathematicalobjectsofarbitrarydimen-sions(hyperplanes)livinginn-dimensionalspace,forarbitrarynaturalnumbern.Thisispresentlyknownasgradedalgebra,wheretherearescalars,vectors,bivectors(kindofantisymmetrictensorsofgradetwo),trivectors,etc.HedidthattenyearsbeforeRiemann’sfamouslecturetotheG¨ottingenfacultyonthespacesthatnowbearhisname,atatimewhereonebarelyhadadditionofvectorsandmultiplicationbyscalars.Highlyrecognizednowadaysalsoishisimplicitcontributiontoprojectivegeometry,withimplicationsfordifferen-tialtopologyandalgebraicgeometry,andwheretheconceptofGrassmannianhonorshisname.Theexteriorproductwasonlythebestknownofthemanydifferentprod-uctsthatheconsidered.Hispaper“OntheVariousTypesofMultiplication”(includedin[43])reportsthreesuchtypes,namelysymmetric,circularandlin-ear.Onewouldthinkthatisbyitselfquiteanachievement,consideringthatthisisinmidnineteenthcentury.Butthenweareforasurprise.Therearesixteendifferentspeciesofsymmetricmultiplications,eightdifferentspeciesofcircularmultiplicationandtwodifferentspeciesoflinearmultiplication,whichhecallsalgebraicandouterrespectively.Notallofthemareequallyimportant.Inthisregard,wefollowCartan’sreportofGrassmann’sproductsinhispa-peroncomplexnumbers[6].Hespeaksofthecomplexandtheinteriorasthemostinterestingamongthecircularproducts.Thealgebraicandtheexteriormultiplicationsarethelinearones.Cartancontinueshisreportspeakingoftwosystemsofnon-algebraicmul-tiplication,orratherasystemwithtwodifferentmultiplications.Thesearetheprogressiveandregressiveproducts,thecombinatorialonesbeenamongtheprogressiveones.Intimatelyrelatedtotheprogressiveproduct,thereistheregressiveproduct,bothofgreatimportanceinmodernalgebra.Cartanalsospeaksinthesamereportofinteriormultiplicationinvolvingeitherprogressiveorregressiveproducts.Ifyouareoverwhelmed,well,thatwasmypurpose.Youhaveexperiencedwhatyouwouldhavefeltifyouhadimmersedhimselfintohiswork.B.2.3TensorandquotientalgebrasNoteverythinginGrassmannworkiscorrectorfinal.Thus,forexample,noattentionisexplicitlygiventowhetheraproductisassociativeornot.Atonepoint,hemaybeassumingitandlaterdropsitalmostwithoutnotice.Thereisalsoanissueastothelimitationsofhisconstrains,whichwillbebetterunderstoodafterwespeakofhisinformalinventionofthegeneraltensoralgebraanditsquotientsalgebras.Inthefirsthalfofpage454of[43](thisisinthepaper“OntheVariousTypesofMultiplication”),Grassmannhasalreadyintroducedtheessentialsof

289B.2.HERMANNGRASSMANN(1808–1877)271theconceptofgeneraltensoralgebra.Ifthereisalimitationinthisdefinition,itistheabsenceofwhatwecalledproperty(c)inourdefinitionoftensorproduct(section7ofchapter3).Thedifferencebetweenthegeneraltensoralgebrasandtheirquotientsalgebrasliespreciselyinthatthelatteronesdonotcomplywiththatproperty.Viewthesecommentsandthosetofollowinthecontextthatwearetalkingabouta1844publication,fourdecadesbeforetheadventofthevectorcalculus.QuotientalgebraswereintroducedbyGrassmannasconstrainsontheprod-uctsofhisgeneral(tensor)algebras.Beawarethatwearespeakingoftensors,nottensorsfields.ThesebecamepartofstructuredmathematicswiththeworkofRicciandLevi-Civita[60].Aspointedoutinaneditorialnoteinthetranslationofthepaper“OntheVariousTypesofMultiplication”,thereisareallimitationintherestrictionofconstraintsbyGrassmanntotwofactors(RecallthattheJacobiidentityisnotofthistype).ItisimportanttonoticethatthedefiningrelationsofCliffordalgebra(seeforinstanceEquations(8.6)and(8.8)ofchapter3areofthistype).ForfurtherdiscussionofthelimitationofGrassmannconstraintsseethateditorialnote.B.2.4ImpactandhistoricalcontextTheimpactofGrassmanninmathematicsgoesfarbeyondalgebraandcalculus.Thus,forinstance,Yaglomreports[93]thatDedekinddevelopedtheaxiomaticdefinitionofthenaturalnumbersfromGrassmann’sconstructions.Thedusthasnotyetsettledonwhatwashisvisiononprojectivegeometry.ThepaperbyDieudonn´eTheTragedyofGrassmann[32]laudshiscontributiontothatgeometry.ItwaseventuallyfollowedbyRotaandcollaboratorspaper[1]whoclaimedthattheBourbakists(Dieudonn´eisoneofthem)hadnotun-derstoodthefullimportofGrassmann’sideasonthesubject.ThatisironicsincebothofthesepartiesmighthavemadeclaimtobeingthegreatestmodernadmirersofGrassmann.Justbeforedying,GrassmanncametothethresholdofCliffordalgebra.Hehadvirtuallyabandonedmathematicsduringtheprevioustenyearsbecauseoffrustrationwiththelackofreceptionofhisideas.TheCliffordproductoftwovectorsisthesumoftheirexteriorandinnerproducts,whichGrassmanncreated.Reportedly,Cartan’sworkisdifficulttounderstand.Grassmannworkismuchmoreso.Thisshouldnotbesurprising.Inadditiontobeingaheadofhistimeandworkingintotalisolationfromacademia,hewasnottrainedintheprevailingmathematicaltraditions.TopmathematicianslikeCauchyandM¨obiuswereputoffbyhisstyle,thusfailingtocapturethetremendousdepthofhisideas.OnehastowonderwhatmighthavebeenGrassmann’slegacyifhehadbeenofferedaprofessorshipatG¨ottingentodevelopandexplainhisideasinamoreconventionalway.Aterrificcross-fertilizationwithGaussandspeciallyRiemannmighthaveensuedbeforethe“horriblevectorcalculus”(Dieudonn´e’s

290272APPENDIXB.“BIOGRAPHIES”(“PUBLI”GRAPHIES)expression),tensorcalculusandgammamatricesemerged.Calculusandgeom-etrymighthavebeenatthebeginningofthe20thcenturybetterthantheyareeventoday.

291AppendixCPUBLICATIONSBYTHEAUTHOROnemainmotivationoftheauthortowritethisbookisthathisworkonunification—whichisnowenteringitshighenergyphaseafterhavingrecentlyenteredthemoregeneralquantummechanicalphase—willnotbeunderstoodwithoutknowledgeofCliffordalgebra,K¨ahlercalculus,Finslerbundlesandhistemporarilyabandonedworkonthegeometrizationofclassicalelectrodynamicswiththosebundles.Thelistofpublicationsthatfollowmaygiveanideaofwherethosetopicsmayhavebeendiscussedingreaterdetail.51.“U(1)×SU(2)fromtheTangentBundle”,J.Physics:ConferenceSeries474,012032(2013).50.“RealUnitsImaginaryinK¨ahler’sQuantumMechanics”.ArXiv-1207.5718vI.49.“RealVersionofCalculusofComplexVariable:Cauchy’sPointofview”.ArXiv1205.4256.48.“RealVersionofCalculusofComplexVariable:WeierstrassPointofView”.ArXiv1205.4657.47.Book“DifferentialFormsforCartan-KleinGeometry”,Abramis,London(2012).46.“Opera’sNeutrinosandtheRobertsonTestTheoryoftheLorentzTrans-formations”,ArXiv1111.2271v2.45.“FromCliffordthroughCartantoK¨ahler”,Hypercomplexnumbersingeometryandphysics,#1(13)Tom7,Moscow,pp.165-180(2010).Itcanbefoundin:http://hypercomplex.xpsweb.com/page.php?lang=ru&id=569.ClickRussianpdf.Scrolldowntopage165.PapersareinRussian,exceptmine,whichisinEnglish.44.“TheFoundationsofQuantumMechanicsandtheEvolutionoftheCartan-K¨ahlerCalculus”.Found.Phys.38,610-647(2008).43.“KleinGeometries,LieDifferentiationandSpin”,DifferentialGeometry–DynamicalSystems10,300-310(2008).273

292274APPENDIXC.PUBLICATIONSBYTHEAUTHOR42.“TheKaehler-DiracEquationwithNon-Scalar-ValuedDifferentialForm”.Adv.Appl.CliffordAlg.18,1007-1021(2008).41.“NewPerspectivesontheKaehlerCalculusandWaveFunctions”.Adv.Appl.Cliffordalg.18,993-1006(2008).40.“RecentDevelopmentsontheFoundationsofClassicalDifferentialGe-ometrywithImplicationsfortheTestingandUnderstandingofFlatSpacetimePhysics”,ProceedingsofXIIIInternationalScientificMeetingonPhysicalInter-pretationsofRelativityTheory,BaumannStateTechnicalUniversity,Moscow(2007).39.“Affinetorsion`alaCartan”,AnnalesdelaFondationLouisdeBroglie32,409-423(2007).38.“TheIdiosyncrasiesofAnticipationinDemiurgicPhyssicalUnificationwithTeleparallelism”,withD.G.Torr.InternationalJournalofComputingAnticipatorySystems19,210-225(2006).37.“AnticipationattheUnificationofGeometryandCalculus”,withD.G.Torr.InternationalJournalofComputingAnticipatorySystems19,194-209(2006).36.“OfFinslerFiberBundlesandtheEvolutionoftheCalculus”,withD.G.Torr.BalkanGeometrySocietyProceedings,GeometryBalkanPress,Bucharest,183-191(2006).www.mathem.pub.ro/dept/confer05/M-VAO.PDF.35.“ADifferentLineofEvolutionofEvolutionofGeometryonMani-foldsEndowedwithPseudo-RiemannianMetricsofLorentzianSignature”,withD.G.Torr.BalkanGeometrySocietyProceedings,GeometryBalkanPress,Bucharest,173-182(2006).www.mathem.pub.ro/dept/confer05/M-VAA.PDF.34.“AGravitationalExperimentInvolvingInhomogeneousElectricFields”,withT.DattaandM.Yin.Proceedingsofthe2004SpaceTechnologyandap-plicationsInternationalForum(STAIF),1214-1221(2004).33.“IsElectromagneticGravityControlPossible?”,withD.G.Torr,Pro-ceedingsofthe2004SpaceTechnologyandapplicationsInternationalForum(STAIF),206-1213(2004).32.“SynchronizationsversusSimultaneityRelations“,withImplicationsforInterpretationsofQuantumMeasurements”,withD.G.Torr.In“Gravita-tionandCosmology:FromtheHubbleRadiustothePlanckScale”,367-376.Editors:R.L.Amoroso,G.Hunter,M.KafatosandJ.P.Vigier,Kluwer,Boston(2002).31.“FromtheCosmologicalTermtothePlanckConstant”,withD.G.Torr.In“GravitationandCosmology:FromtheHubbleRadiustothePlanckScale”,Kluwer,Boston,1-10(2002).30.“QuantumCliffordAlgebrafromClassicalDifferentialGeometry”,withD.G.Torr.J.Math.Phys.43(3),1353-64(2002).29.“TheCartan-CliftonMethodoftheMovingFrame:FinslerianBundlesonRiemannianDistances”,withD.G.Torr.Algebras,GroupsandGeometries,17(3),361-374(2001).28.“MarriageofCliffordAlgebraandFinslerGeometry:aLineageforUnification”,withD.G.Torr.Int.J.Theor.Phys.40(1),273-296(2001).

293APPENDIXC.PUBLICATIONSBYTHEAUTHOR27527.“Clifford-ValuedClifforms:aGeometricLanguageforDiracEqua-tions”.WithD.G.Torr.InCliffordAlgebrasandTheirApplicationsinMath-ematicalPhysics.;Vol.1(AlgebraandPhysics),pp.143-162.Editors:R.AblamowiczandB.Fauser.Birkh¨auser,Boston(2000).26.“TheTheoryofAccelerationwithinitsContextofDifferentialInvariants:TheRootoftheProblemwithCosmologicalModels?”,withD.G.Torr.Found.Phys.29,1543-1580(1999).25.“TheCartan-EinsteinUnificationwithTeleparallelismandtheDis-crepantMeasurementsofNewton’sConstantG”,withD.G.Torr.Found.Phys.29,145-200(1999).24.“TeleparallelK¨ahlerCalculusforSpacetime”,withD.G.Torr.Found.Phys.28,931-958(1998).23.“TheConstructionofTeleparallelFinslerConnectionsandtheEmer-genceofanAlternativeConceptofMetricCompatibility”,withD.G.Torr.Found.Phys.27,825-843(1997).22.“TheEmergenceofaKaluza-KleinMicrogeometryfromtheInvariantsofOptimallyEuclideanLorentzianConnections”,withD.G.Torr.Found.Phys.27,533-558(1997).21.“ElementaryGeometriesUnderlyingtheTheoryofNon-linearEuclideanConnections”,withD.G.Torr,Cont.Math.,196,301-309(1996).20.“CanonicalConnectionsofFinslerMetricsandFinslerianConnectionsonRiemannianMetrics”,withD.G.Torr,Gen.Rel.Grav.28,451-469(1996).19.“TheCornerstoneRoleoftheTorsioninFinslerianPhysicalTheories”,withD.G.Torr,Gen.Rel.Grav.27,629-644(l995).18.“FinslerianStructures:TheCartan-CliftonMethodoftheMovingFrame”,withD.G.Torr.J.Math.Phys.34(10),4898-4913(1993).17.“GeometrizationofthePhysicswithTeleparallelism(II):TowardsaFullyGeometricDiracEquation”,withD.G.TorrandA.Lecompte.Found.Phys.22,527-547(1992).16.“GeometrizationofthePhysicswithTeleparallelism(I):TheClassicalInteractions”.Found.Phys.22,507-526(1992).15.“ConservationofVector-ValuedFormsandtheQuestionoftheExis-tenceofGravitationalEnergy-MomentuminGeneralRelativity”,withD.G.Torr.Gen.Rel.Grav.23,713-732(1991).14.“OntheGeometrizationofElectrodynamics”.Found.Phys.21,379-401(1991).13.“TheBreakingoftheLorentztransformationsandtheGeometrizationofthePhysics”,withD.G.Torr.Nucl.Phys.B.(Proc.Suppl.)6,115(1989).12.“OnTestingtheLineElementofSpecialRelativitywithRotatingSys-tems”,withD.G.Torr.Phys.Rev.A39,2878(1989).11.“ElectrodynamicsoftheMaxwell-LorentzTypeintheTen-DimensionalSpaceoftheTestingofSpecialRelativity:ACaseforFinslerConnections”,withD.G.Torr.Found.Phys.19,269(1989).10.“RevisedRobertson’sTestTheoryofSpecialRelativity:SupergroupsandSuperspace”.Found.Phys.16,1231(1986).

294276APPENDIXC.PUBLICATIONSBYTHEAUTHOR9.“RevisedRobertson’sRestTheoryofSpecialRelativity:Space-TimestructureandDynamics”,withD.G.Torr.Found.Phys.16,1089(1986).8.“KinematicalandGravitationalAnalysisoftheRocket-BorneClockEx-perimentbyVessotandLevineUsingtheRevisedRobertson’sTestTheoryofSpecialRelativity”.Found.Phys.16,1003(1986).7.“RevisedRobertson’sTestTheoryofSpecialRelativity”.Found.Phys.14,625(1984).6.“ProblemsofSynchronizationinSpecialRelativity:(AReplytoG.CavalleriandG.Spinelli)”,J.G.Vargas.Found.Phys.13,1231,(1983).5.“NonrelativisticPara-MaxwellianElectrodynamicswithPreferredRef-erenceFrameintheUniverse”.Found.Phys.12,889(1982).4.“SpontaneousParaLorentzianConserved-VectorandNonconserved-AxialWeakCurrents”.Found.Phys.12,765(1982).3.“NonrelativisticPara-LorentzianMechanics”.Found.Phys.11,235(1981).2.“RelativisticExperimentswithSignalsonaClosedPath(ReplytoPod-laha)”.Lett.NuovoCimento28,289(1980).1.“CommentonLorentzTransformationsfromtheFirstPostulate”.Am.J.Phys.44,999(1976).

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302May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

303IndexCrcompatible,39Barnabei,262Crdifferentiablemanifold,39basemanifold,91,103,151n-volume,38baseoftheframebundle,87r−,32basesofvaluedness,150“canonical”KKspace,219basisofavectorspace,4510-dimensionalPoincar´egroup,102basisofdifferential1−forms,29basisofdifferential2−forms,33absolutesimultaneity,219bendinginvariant,257action,28Bianchiidentity,13,159,165actionofaLieoperator,143bivector,23,63adaptedtosurfaces,248,251boostsofparticles,220affinecurvature,14,23,90,100,101,boundaryofanopencurve,77149,168Brandt,217affineextension,12,102,112,178Brini,262affineframebundle,97,174Bryant,263affineframes,87affinegroup,85,87calculusofcomplexvariable,240affineparameters,158calculusofvector-valueddifferentialforms,affinespace,16,8544affinetransformation,168,192canonicalframefield,259affine-linearconnection,168canonicalframefieldofasurface,254,algebra,16255analytic,240canonicallydetermined,248anholonomic,29Cartancalculus,67anholonomicbasisfields,95Cartangeometry,132annulmentofconnection,154,189,199Cartan’sgeneralization,107antiparticles,228,241,243,244Cartan’smethodofequivalence,195associatedvectorspace,85Cartan-Einsteinunification,266associativeproperty,70Cartan-Finslerconnection,217,265atlas,39Cartan-K¨ahlertheory,228,263autoparallel,135,158,185Cartaniangeneralization,90autoparallelsinFinslerframebundles,Cartesiancoordinates,38,74219Cartesianform,54auxiliarybundles,223,242–244Cauchy,225,271Cauchycalculus,240bacheloralgebra,10Cauchy’stheory,240ball,4Cauchy-Riemannconditions,240285

304286INDEXchangeofbasisofdifferentialn−forms,conservationlawingeneralrelativity,38164changeofbasisofdifferential1−forms,conservationofcharge,7935conservationofthecurvature,201changesofbasis,46conservationofthetorsion,163charge,245constantbasisfields,99,162chargedensity,79constantframefields,248chart,24,39constantsections,110Chern,217,263constanttensorfield,163choiceofbasisfield,154contorsion,178,179,188Christoffel,140,194contourintegral,240Christoffelsymbols,142,174contravariant,23circumflex,117,121contravariantvector-valuedness,100class,38contravariantvectors,114classofthemanifold,40conventionalityofsynchronizations,219convex,72classicaldifferentialgeometry,4coordinatebasisfield,56,92Clifford,226coordinatefunctions,12,21,24,39Cliffordalgebra,9,16,61,63–65,81,coordinateline,86205,237coordinatemap,24Cliffordalgebraofspacetime,65coordinatepatches,39,94Cliffordproduct,63,64coordinatesurface,259Cliffordvaluedness,244coordinatesystem,24,80clifforms,64coordinatetransformations,39Clifton,217coordinatesinthefibers,87closedcurves,14cotangentvaluedness,44co-derivative,238cotangentvectorspaces,48cochain,6,10,30cotangentvectors,44,114Codazziequations,257covariant,23,116Columbus,134,135covariantderivative,13,99,108,162,Columbusconnection,129,130,132,165134,135,137–139,265covariantvector-valuedness,100Columbusrule,134,135covariantvectors,114componentsoftheconnection,96crossproduct,30,31componentsofthetorsion,180Crowe,227conceptofgeometricequality,193curl,73,238connection,4,23curldifferentialform,76connectioncoefficients,96curlvectorfield,76connectiondependent,116currentdifferential3−form,79connectiondifferentialforms,89currentofcharge,202connectionequations,89curvature,91,247connectionofthetorus,138curvaturetensor,23connection-dependent,100curveism,11connection-independent,100curves,247conservation,159curvilinearcoordinatesystems,92conservationlaw,13,77–79,161,162curvilinearquadrilateral,134,154,157

305INDEX287cyclicproperty,14Einstein’ssynchronizationprocedure,cylindricalcoordinates,75219Einstein’stensor,198,201–203,205,Dedekind,271207Deicke,217Einstein’stheoryofgravity,34,211demiurge,266Einstein’sthesisoflogicalhomogene-densities,74ity,246derivedconcepts,242Einstein’sviewofparticles,241Dieudonn´e,225,267,268,271electromagneticdifferential2−form,37,differenceinviewpoint,19456,187differentiablefunctionsonmanifolds,electromagneticenergy-momentum3−form,39208differentiablemanifold,41electromagneticfield,10differential1−form,21,25,26electromagneticinteraction,245differential2−forms,31electromagneticLagrangian,8differentialr−form,32,68electrons,245differentialform,5,21,23elementarygeometry,217differentialrotation,218embedded,247differentialtranslation,218emergenceofspin,241dimensionofthevectorspace,45energy,28,245Dirac,265energydensity,8Diracequation,241energyoperator,221Diractheory,241energy-momentum,8,13,14Dirac’scalculus,228,240,241energy-momentum(four)vector,202directionofmaximumchange,76energy-momentumdifferentialform,202directionsofprincipalcurvature,255energy-momentumtensor,198,202discontinuityinthetangent,258entangledabinitio,244displacements,192equalityoftwor−forms,36distance,112equalityofvectors,13distanceoncurves,249,252equationsofstructure,90,91,147divergence,73,74,76,238equivalenceofcurves,146domainoftheparameters,248equivalenceproblem,197Donaldson,17Erlangenprogram,192dualbases,47Euclideanalgebra,17dualvectorspace,43,47Euclideanbases,109Euclideanconnection,10,12,16,133,egregiumtheorema,257135,171,175,176,197Ehresmann,265Euclideancurvature,133,176Einstein,6,8,10,194,203,221,222,Euclideanframe,109,172266Euclideanframebundle,109,174Einsteindifferential,3−form,203,205,Euclideangroup,109207,209Euclideanplane,131Einsteinelevators,11Euclideanpointspace,109Einsteinsummationconvention,22,24,Euclideanspace,16,10946Euclideanvectorcalculus,63Einstein’sequations,202,208Euclideanvectorspace,48

306288INDEXEuclidean-Kleingeometry,175foundationsofphysics,6,7Euler’stheorem,256four-potentialform,29Euler-Lagrangeequations,186framebundle,87,88extension,174freeparticleenergy-momentumform,extensionoftheCartancalculus,24329extensionofthemetric,113FrenchAcademy,195exteriorcalculus,5,63Frenetequations,253exteriorcovariantderivative,13,115,Frenetframefield,248161frequency,28exteriorderivative,13,67–69frequency-wave-numberform,29exteriordifferential,68Frobeniustestofintegrability,122,141exteriorproduct,22,30,32,35,63,70,Frobeniustheorem,90,148,168237functionofcurves,115,156exteriorproductofforms,69functional,15exteriorvalueddifferentialforms,221functions,15extremals,186fundamentalgroup,192failuretoclose,138G¨ottingen,225,271falsespacesofRiemann,11,171,193,Gardner,263,264,267,268195,266Gauss,225,251,255,271familiesofcurves,157Gauss-Bonnet,258,262Faraday’slaw,79Gauss-Bonnettheorem,247farewelltoauxiliarybundles,246generalaffinetransformation,104Feynman,8generallineargroup,47,87fiber,87,151generalrelativity,13,201,221fiberbundle,88generaltensoralgebra,57fields,15generalizedenergy-momentumdifferen-fifthdimension,218tialform,29fifthdirection,220geodesiccurvatures,253finiteLiealgebras,242geodesiccurves,157Finslerbundle,187,216geodesicdeviation,165,199Finslerframebundle,218,221geodesicframefield,253,258Finslergeometry,11,217geodesictorsionofthecurve,254Finslerspaces,11geometricequality,129Finslerianaffineframebundles,217geometricinterpretation,156Finslerianbasemanifold,220geometricpedigree,242Finslerianfibration,102geometricsignificanceoftheBianchiFinslerianrefibration,103identities,161Finsleriantorsions,188geometry,16firstfundamentalform,249geometrybecomescalculus,246fivedimensionalspace,218Gibbs,227Flanders,115globalresult,258flowofavectorfield,144Goldschmidt,263Foldy-Wouthuysen,241GR,7,8Foldy-Wouthuysentransformations,228,grade,61241,243gradedalgebra,25,242,270

307INDEX289gradient,73infinitegroupofdiffeomorphisms,192,gradientdifferentialform,29,74,76242Grassmann,215,227,262,263,269–infiniteLiegroup,192,242271infinitelysmallcontour,160,161Grassmann’sOpusMagnum,225infinitesimaltransformation,143Grassmann’sproducts,270infinitesimaltranslations,193Grassmannian,270inspection,180gravitation,203integrability,90,148gravitationalenergy,8integrabilityconditions,90,159gravitationalenergy-momentum,202interior“derivative”,238gravitationalinteraction,11interiorderivative,76Greekindices,172internalproperty,244Griffiths,263internalsymmetries,244groupofmatrices,80invariant,99,100groupofrotations,81invariantforms,107,111groupoftransformations,80invariantofthemetric,194groupoftranslations,81invariantoperator,144,145inverseofavector,66Hamilton,226,227isotropiccoordinates,182harmonic,119issuesofintegrability,243heattransfer,195Ives-Stilwell,219Heaviside,227Hestenes,9Jacobiidentity,271Higgsfield,9Jacobian,31Higgsparticle,9highenergyphysics,241K¨ahler,241Hilbert,262K¨ahleralgebra,64,124Hodgedual,61,120,203K¨ahlercalculus,9,124,125,220,222,Hodgeduality,116,238243holomorphic,240K¨ahlerdifferentiation,220holonomic,149K¨ahlerequation,228,241,243holonomicbases,91,150K¨ahlerviewofquantummechanics,222holonomicbasisfields,93K¨ahler’scalculus,240holonomicsections,94,98K¨ahler-Diracequation,228horizontal,147,197Kaluza-Klein,265horizontalpart,107,108,258Kaluza-Kleinextension,216horizontality,107,108,148,151,168,Kaluza-Kleinspace,244196Kennedy-Thorndike,219horriblevectorcalculus,271Klein,215,269hyperplane,130,270Kleingeometry,16,163,191,222,242Kroneckerdelta,22ideal,15,66,223Kronheimer,17idempotent,62,66,221,223identityfunction,24Lagrangianapproach,8identitymap,39Landau,8,264imaginarypart,240Laplacian,73,119,120

308290INDEXLatinindices,172magnitude,49LCC,8,10,129,130,138,139map,15LecturesonIntegralInvariants,227mass,245leftideal,221mathematicalvirus,9leftinvariantforms,89,168matrices,45Leibnizrule,99,238Maurer-Cartanequations,105lengthofcurves,185Maurer-Cartandifferentialform,80Levi-Civita,137,142,192,194,242,maximalatlas,39261Maxwell’sequations,10,208,265Levi-Civitaconnection,12,13,119,142,Mead,7176,197meridians,135,138Levi-Civitaconnectionofthesurface,methodofequivalence,194256metric,10,75,111,173Levi-Civitacurvatureofthesurface,257metricasaderivedinvariant,175Levi-Civitasymbols,177metriccompatibility,150Levi-Civitatensor,61metriccurvature,10,14Lichnerowiczonthetensorcalculus,44,metricstructure,7458,230Meusnier’stheorem,256Lie,269Michelson-Morley,219Liealgebra,79,80,103,104,242modules,15,25Liealgebravalued,13modulesof1−forms,88Liebrackets,80momentum,28Liederivative,9,143–145,241monocurvaturevirus,10Liedifferentiation,143,241monomial,23Liegroup,79,104movingframemethod,248,250Lieoperator,145multivectors,64Lieproduct,80mutuallyannullingprimitiveidempo-Lifshitz,8,264tents,223lineartransformation,81linearlyindependentr−forms,33naturalconnection,12linesofconstantdirection,132,135,naturalliftingcondition,245137,138naturalliftings,188linesofcurvature,255negativeenergies,241Lipshitz,140,194negativeenergysolutions,9localconcepts,247NewYork,136localcoordinates,24non-Euclideanenvironment,171logicalhomogeneity,6non-holonomic,97,149Lorentzforce,187,218non-holonomicbasis,29Lorentzmetric,245non-holonomicbasisfields,93Lorentzspace,181non-holonomicframefield,95Lorentztransformations,245non-holonomicsection,96Lorentz-Finslergeometry,218non-holonomictangentbasisfields,98Lorentziansignature,112non-horizontal,148non-tensorialtransformation,178M¨obius,271norm,49Madrid,136normalcomponent,248

309INDEX291normalcoordinates,199,200primordialfield,244normaltoasurface,250principalcurvatures,260notionofparallelism,193principalfiberbundles,88notionsofaEuclideannature,192problemofequivalence,141,194,195,197,229oldRiemanniangeometry,194projectionmap,92,97,146orbitalangularmomentum,9,243projectivegeometry,192orientationofspin,244projectivegroup,192orientedcurve,26properRiemannianspace,197orthogonalcoordinatesystems,76properlyEuclidean,49,174orthogonalization,52pseudo-Euclidean,172orthonormalbasis,49pseudo-Euclideanspace,112orthonormalbasisfields,75,76pseudo-Euclideanvectorspace,49osculatingplane,253pseudo-orthonormalbases,49,173overdot,186pseudo-spaces,191pull-back,71,92,97parallelprojections,86pull-backofadifferentialform,71paralleltransport,155pull-backtothesurface,250paralleltransporting,137pull-backstofibers,107,147parallels,135,138pull-backstosections,94parametrizations,248push-forward,93,97,168ParisianAcademy,195passiveoperator,146quadraticsymmetricdifferentialform,Pauli,19423,140Peano,226quantummechanics,221Pfaffian,106quantumphysics,203phaseofawave,28quarks,242,246phaseshifts,221quasiquadrilateral,161Pl¨uckeriancoordinates,218quotientalgebra,12,58,271plane,130planecurves,253radiusofcurvature,253Poincar´e,267rank,43,55,59,60Poincar´egroup,245reciprocalbasis,50,76Poincar´esymmetry,244rectilinearcoordinates,86pointofmassproblem,208referencesystem,86polarcoordinates,24,93region,24polarmetric,181Reissner-Nordstromproblem,208pole,39,240relationofequality,135polygons,40,258relativisticquantummechanics,63,125,polynomials,45219,228positrons,245restriction,121,173,174,217,228postPauli-Dirac,241restrictionsofaffineconnections,172Poyntingvector,8rhumblines,135,138preferredframe,244,245Ricci,194,242,261,267primedquantities,96Riccitensor,198,201primordialconcept,242,244Riemann,13,140,199,215,226,271

310292INDEXRiemannplustorsion,171smallnessofcurves,160Riemann’scurvature,14,129,141,176,smoothedpolygon,258203smoothing,258Riemann’sequivalenceproblem,140solderingforms,168Riemanniangeometry,14,242space-timemanifold,27Riemannitisvirus,11specialpoint,12ring,14sphere,4Rodriguesequation,256sphericalcoordinates,32,33,75Rodrigues’estheorem,256sphericalmetric,182roleofgrouptheory,191spin,9,14,145,228,243,245Rota,226,262,271spinorbundles,88Rudin,6,10,26,115spinorsolutions,241rulesofthumb,181spinors,62,66,243standardfibration,102scalarproduct,48stationarylength,185scalarvalueddifferentialforms,5Stern-Gerlachexperiment,244scalarvaluedfunction,69Sternberg,268scalar-valueddifferential0−form,22Stokestheorem,68,69,71,72scalar-valuedness,243strictlyharmonic,119Schmeikal,223structureconstants,106Schmidt’smethod,52,53Struik,253,258Schmidt’sorthogonalizationprocedure,Study,26753subalgebras,58schoolofHamburg,192subgroupoftranslations,103Schouten,192subspace,45Schwarzschildmetric,182,208surface,4,247secondcovariantderivatives,101surface-adapted,248secondderivative,70secondfundamentalform,248,254T¨ubingen,269secondranktensors,59tangent,146section,91tangentplane,80,250sectionofthebundle,91,96tangenttensors,43section’sperspective,97tangentvector,43,146signofenergy,243tangentvectorspaces,43,48signature,78,112teleparallelconnection,266simpler−form,32teleparallelequivalent,13Singer,268teleparallelism,8,13,130,160,163,skew-symmetric,43245,266skew-symmetricmultilinearfunctions,tensoralgebra,24230tensorcurvature,13skew-symmetricpart,36tensorfield,60skew-symmetricpartoftheconnection,tensorproduct,43,58,60,173243tensorproductofvectorspaces,56skew-symmetry,13,32tensorvaluedness,23Slebodzinski,143,144,241,266tensor-valueddifferentialforms,21,58smallcomponents,241tensorialcharacter,152

311INDEX293tensoriality,151VanStockum’smetric,183tensors,59variationalprinciple,28theoremofresidues,239,240varioustypesofmultiplication,270thermodynamicalsystem,27vectorbundles,88thesisoflogicalhomogeneity,222vectorfield,56,143,145tinycurves,153vectorfieldscomponents,56topologicalinvariant,258vectorproduct,62,237topologicalissue,130vectorspace,15,45topologicalresult,258vectorvalued,5torsion,14,23,90,91,100,101,138,vector-valueddifferential1−form,22,139,149,150,160,168,171,88247vector-valueddifferentialform,22,99torsionofthecurve,253verticalpart,258torus,12volumes,37totalcurvature,255transformationproperties,242wave-numbers,28transformations,87weakandstronginteractions,222transformationsinthefiber,102Weingarten,251translation,104,191Weingartenequations,252transmutationvirus,10Weitzenbock,266,267tripleofgroups,217Weyl,262,264,267trivector,64Whitney,41truerestenergy,241Yaglom,271typesofindices,12Yang-Mills,103,221umbilics,255Yang-Millsconnection,222unisexvirus,10Yang-Millsdifferentialgeometry,58unitimaginary,221,244Yang-Millstheory,218,223,242unitnormal,250zerotorsion,164valuedness,6zero-form,69VanDantzig,265