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1、AnnalsofMathematicsOntheCurvaturaIntegrainaRiemannianManifoldAuthor(s):Shiing-shenChernReviewedwork(s):Source:TheAnnalsofMathematics,SecondSeries,Vol.46,No.4(Oct.,1945),pp.674-684Publishedby:AnnalsofMathematicsStableURL:http://www.jstor.org/stable/1969203.Accessed:26
2、/07/201200:19YouruseoftheJSTORarchiveindicatesyouracceptanceoftheTerms&ConditionsofUse,availableat.http://www.jstor.org/page/info/about/policies/terms.jsp.JSTORisanot-for-profitservicethathelpsscholars,researchers,andstudentsdiscover,use,andbuilduponawiderangeofconte
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4、ofMathematics.http://www.jstor.orgANNALSOFMATHEMATICSVol.46,No.4,October,1945ONTHECURVATURAINTEGRAINARIEMANNIANMANIFOLDBYSHIING-SHENCHERN(ReceivedMay23,1945)IntroductionInapreviouspaper[1]wehavegivenanintrinsicproofoftheformulaofAllendoerfer-WeilwhichgeneralizestoRie
5、mannianmanifoldsofndimensionstheclassicalformulaofGauss-Bonnetforn=2.Themainideaoftheproofistodrawintoconsiderationthemanifoldofunittangentvectorswhichisintrinsi-callyassociatedtotheRiemannianmanifold.DenotingbyR'theRiemannianmanifoldofdimensionnandbyM2"'themanifoldo
6、fdimension2n-1ofitsunittangentvectors,ourproofhasled,inthecasethatniseven,toanintrinsicdifferentialformofdegreen-1(whichwedenotedbyII)inM2"'.WeshallintroduceinthispaperadifferentialformofthesamenatureforbothevenandodddimensionalRiemannianmanifolds.Wefindthatthisdiffe
7、rentialformbearsacloserelationtothe"CurvaturaIntegra"ofasubmanifoldinaRieman-nianmanifold,becauseitwillbeprovedthatitsintegraloveraclosedsubmani-foldofR'isequaltotheEuler-Poincarecharacteristicofthesubmanifold.Themethodcanbecarriedovertodeducerelationsbetweenrelative
8、topologicalinvariantsofasubmanifoldofthemanifoldanddifferentialinvariantsderivedfromtheimbedding,andsomeremarksaretobeaddedtothisef
