S^n+1中具有调和Mobius曲率张量的超曲面

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1、维普资讯http://www.cqvip.com第37卷第1期数学进展、1．37．No．12oos~2月ADVANCESINMATHEMATICSFeb．，2008HypersurfacesWithHarmonicM6biusCurvaturein叶lLITongzhu，SUNHuafei(DepartmentofMathematicsBeoingInstituteofTechnology，Bering，100871P．R．China)Abstract：Letz：』n．_÷Sn+beahypersurfaceinthefn+11．dimensionalu

2、nitsphereSwithoutumbilics．TWObasisinvariantsofzundertheM6biustransformationgroupofSatetheM5biusmetricgandtheM5biussecondfundamentalformB．whichdeterminethehypersurfaceuptoaM5biustransformationifn3．Inaddition，theMSbiusformiSaimportantinvariant．Theassumption西：0iSnecessaryinsomeclass

3、ificationtheoremsInthispaper，weconsiderthen1dimensionalhypersurfacesfn>31withvanishingM5biusform西．weclassifythehypersurfaceswithharmonicM5biuscurvaturetensor．Moreover．weclassifyallEinsteinhypersurfacesandallhypersurfacesofconstantsectionalcurvaturewithrespecttoM6biusmetric．Keywor

4、ds：M5biusgeometry；harmonicM5biUScurvaturetensor；EinsteinmanifoldMRf2000)SubjectClassification：53A03／CLCnumber：0186Documentcode：AArticleID：1000．0917(2008)01．0057-100IntroductionLet：Mn__+Sn+beahypersurfaceinthe(几+1)．dimensionalunitsphereSn+withoutumbilics．Let{et)bealocalorthonormal

5、basisforthefirstfundamentalformI=dx．dxwithdualbasis)-LetII=∑巧hij8i8jbethesecondfundamentalformofandH=元1∑thiithemeancurvatureof．WedefineP2n(j一n日)，whereJJ_Jjisthenormwithrespecttotheinducedmetricdx·dxonM．InWangintroducedtheM6biusmetricg=p2dx·dx，theMSbiusform=p∑i，theBlaschketensorA=

6、p2∑巧AijOiOjandtheM6biussecondfundamentalformB=p∑巧BijOiOj，where=一p一e(日)+E(hij-HSij)e，(1。gp))，Aij-p。。(Hessij(1ogp)一ei(1ogp)ej(1ogp)一Hh巧)1(IIV1。gpI一一1+日)‰Bijp一(一日)．Receireddate：2006-06．21_Foundationitem：ThisworkissupportedbytheTianyuanFoundationofMathematicsE-maihlitz@pku．org．cn维普资讯

7、http://www.cqvip.com数学进展37卷whereHessijandVaretheHessian-matrixandthegradiantwithrespecttod‘dz．ThehypersurfacesinS+withoutumbilicsarecompletelydeterminedbyitsMSbiusmetricganditsM6binssecondfundamentalformBwhenn3(cf．[8】)．Tostateourresults，werecallthatfor(n+1)一dimensionalhyperbolics

8、paceofconstantsectionalcurvature—c<0，H+(