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ID:33937419
大小:1.29 MB
页数:14页
时间:2019-03-02
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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
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