资源描述：
《on curvature and characteristic classed of a remannian manifold.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、OnCurvatureandCharacteristicClasseso
2、aRiemannManifoldVonSn~NG-SHV.NCHEI~inChicagoDedicatedtoWILHELMBLAS(~ttKEonHisSeventiethBirthday1.IntroductionThedevelopmentinrecentyearsofthetheoryoffiberbundlesindifferentialgeometryhasledtorelationsbetweenthecurvatureofacompactRIwMANNmanifoldandcert
3、ainglobalinvariants,theso-calledcharacteristicclasses.ThesimplestamongtheserelationsistheGAuss-BO~N~,Tformula1),whichexpressestheEULER-PoI~c~a~characteristicofacompactorientableI~I~M~NmanifoldasanintegralinvolvingthecomponentsoftheI~IEMA_,NTN-CHttISTOFFELcurvaturetensor.Thereareotherrela
4、tions,tobegivenbelow,betweenthePON~AGINcharacteristicclassesanddifferentialformsconstructedfromthecurvaturetensor.Thepurposeofourpaperistwofold:Toexpresstheserelationsinaconvenientformandtodrawsomeconclusionsfromthem.ThedifferentialformswhichexpresstheEULE~-PoI~cA~characteristicandthePO~
5、TRZAGI~classesbearacloserelationshipwithskew-symmetricdeterminantsandPF~TI~functions.Atheoremconcerningthem,whichwewillfinduseful,isgiveninw2.Inw3weshallintroducenotionsbywhichourrelationscanbebestformulated.Inw4wederivesomeimpli-cationsofthecurvaturepropertiesonthecharacteristicclasses.
6、Inparti-cular,wegiveaproofofatheoremofJ.MIT,NOR~)tOtheeffectthatanorientablecompacttCI~,M~_~NmanifoldoffourdimensionshaspositiveEULE~-PoINCAR~characteristic,ifitssectionalcurvatureisalwayspositiveoralwaysnegative.Amongtheimplicationsoftheserelationsisthefactthattheyprovideawayforeffectiv
7、ecomputationofthecharacteristicclasses.Forinstance,itwillbeeasytocomputethePO~TVZAGI~classesofthecomplexpro-jectivespace,usingitsellipticttermitianmetric.Althoughwewillnotcarrythisout,weshallmakeanapplicationoftheknowledgeoftheseclassesbyderivingthefollowingtheorem:Thecomplexprojectivesp
8、aceP~of(complex)dimensionmcannotbedifferentiablyimbeddedinarealEuclideanspaceofdimension3m~1or3m~2,accordingasmisevenorodd.~)Cf.,forinstance,CHE~tN,Topicsindifferentialgeometry,pp.38~41,Insti-tuteforAdvancedStudy,Princeton,USA.ThismonographwillbequotedasTDG.2)Iowethistoan
