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1、AfurthermethodinglobaldifferentialgeometryByUDoSmo~DedicatedtoE.SPERNERonhisseventiethbirthdayAbstract.Weinvestigate(0,2)4ensors,whichfulfilCodazzi-equations,onclosedRiemannianmanifoldswithnonnegativesectionalcurvature,andgivevariousapplicationsinglobaldifferentialgeom
2、etry.A.IntroductionandbasicresultsThereareseveralmethodsinglobaldifferentialgeometryofsub-manifoldstoproveuniqueness-theoremsandrelatedresults:theindex-method,theapplicationofthemaximumprinciple,theapplicationofmethodsofthetheoryofholomorphicandmeromorphicfunctions,esp
3、eciallyinthetheoryofminimalsurfaces,andthemethodofintegralformulas.Themethodwhichwewillstudyinthefollowingcanbeappliedaswellastolocalastoglobalproblems(inthecompactcasewewilluseintegralformula~too).Atypicalstepoftheproofistoshowthatacertaintensoriscovariantconstant.Int
4、hecaseofthesecondfunda-mentalformcomputationsofthistypeweremadebyseveralauthors(SIMONS[17];doCARMO-CHERI~-KOBAYASKI[1],NomzvandSMYTH[28]):theycalculatedthesquareofthenormofthesecondfundamentalformtensor.MoregeneralcomputationsweremadebyH.F.M~z~.RandD.SII~GLV.Y([9],[26]
5、).Theresultsof~O~ZN]~Rarenotyetpublished(theyweregiveninalectureattheGeometrie-TagungOberwoffach1972).Wewillprovesomeoftheirresultsagainunderweakerassumptions.TosketchsomeofthebasicideaswewillproveasanexamplethewellknownclassicaltheoremofChristoffel(Thefollowingproofis
6、alsoknowntoH.F.Mi~Z~ER,asIknowfromtheGeometrie-TagungOberwoffaeh1972).Theorem.Letx,x*:S"-->E8beembeddingso]thetwo-sphereintothethree-dimensionalEuclideanspacesothatx(M),x*(M)areovaloidswithpositiveGaussiancurvatures.Letx*.x-1:x{M)--->x*(M)beamappingAfurthermethodinglob
7、aldifferentialgeometry53ofparallelnormalsandletbeR1+R2=R*+R*atcorreslaondingtaoints(R~,R*arethe1orincipalradiiofcurvature).Thenx(M)andx*(M)arecongruent.Proof.Takethecommonthirdfundamentalform111=111"=e~jdu~dujasmetricon$2;letb~j,bEbethetensorsofthesecondfundamentalform
8、s//,//*;asbothfulfilCodazzi-equationswithrespecttoIII,sodoesthedifferencetensorA~j:=b~--b~,andtraceA=R1--R*q-R~/~*=0.
