Differential Geometry its past and its future.PDF

《Differential Geometry its past and its future.PDF》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、Actes,Congrèsintern,math.,1970.Tome1,p.41à53.DIFFERENTIALGEOMETRY;ITSPASTANDITSFUTUREbySHIING-SHENCHERN(*)A.Introduction.Itwasalmostacenturyago,in1872,thatFelixKleinformulatedhisErlangerPro•gram.Theideaofunifyingthegeometriesunderthegroupconceptissimpleandattractiveanditsapplicationsinth

2、ederivationofdifferentgeometricaltheoremsfromthesamegroup-theoreticargumentareusuallyofgreatelegance.Thisleadstothedevelopmentofdifferentialgeometriesofsubmanifoldsinhomogeneous(orKlein)spaces:conformai,affine,andprojectivedifferentialgeometries.Thelatterhadinparticularanenergeticdevelop

3、mentinthetwenties.ItwasalsoaboutacenturyagothatthegreatestmoderndifferentialgeometerElieCartanwasborn(onApril9,1869).AmonghiscontributionsofabasicnaturearehissystematicuseoftheexteriorcalculusandhisclarificationoftheglobaltheoryofLiegroups.FiberspacesalsofindtheirorigininCartan'swork.Dif

4、ferentialgeometryisthestudyofgeometrybythemethodsofinfinitesimalcalculusoranalysis.Amongmathematicaldisciplinesitisprobablytheleastunderstood(1).Manymathematiciansfeelthereisnogeometrybeyondtwoandthreedimensions.Theadventintohigherandeveninfinitelymanydimensionsdoesmaketheintuitionunreli

5、ableandthedependenceonalgebraandanalysismandatory.Thebasisofalgebraisthealgebraicoperationsandthebasisofanalysisisthetopologicalstructure.IwouldliketosurmisethatthecoreofdifferentialgeometryistheRiemannianstructure(initsbroadsense).Themainobjectofstudyindifferentialgeometryis,atleastfort

6、hemoment,thedifferentiablemanifolds,structuresonthemanifolds(Riemannian,complex,orother),andtheiradmissiblemappings.Onamanifoldthecoordinatesarevalidonlylocallyanddonothaveageometricalmeaningthemselves.Historicallythedifficultyinachievingaproperunderstandingofthissituationmusthavebeentre

7、mendous(IwonderwhetherthiswaspartofthereasonwhichcausedHadamardtoadmithis(*)ThispaperwaswrittenwhentheauthorheldaResearchProfessorshipoftheMillerInstituteandwasunderpartialsupportofNSFgrantGP20096.(*)G.D.BIRKHOFF,«Thesecondisadisturbingsecretfearthatgeometrymayultimatelyt