S^n+1中具有调和Mobius曲率张量的超曲面.pdf

维普资讯 第37卷第1期 数 学 进 展 、 1．37．No．1 2oos~2月 ADVANCESIN M ATHEM ATICS Feb．，2008 HypersurfacesW ith H armonicM 6biusCurvaturein 叶l LITongzhu ， SUN Huafei (DepartmentofMathematicsBeoingInstituteofTechnology，Bering，100871P．R．China) Abstract：Letz：』n．_÷Sn+ beahypersurfaceinthefn+11．dimensionalunitsphere S withoutumbilics．TWObasisinvariantsofzundertheM6biustransformationgroupof S atetheM5biusmetricgandtheM5biussecondfundamentalform B．whichdetermine thehypersurface up to aM5biustransformation ifn 3 ． Inaddition，theMSbiusform iSaimportantinvariant．Theassumption 西 ：0iSnecessary insomeclassificationtheorems Inthispaper，weconsiderthen1dimensionalhypersurfacesfn 31withvanishingM5bius form 西．w eclassifythehypersurfaceswithharmonicM5biuscurvaturetensor ． Moreover．we classifyallEinsteinhypersurfacesand allhypersurfacesofconstantsectionalcurvaturewith respecttoM6biusmetric． Keywords：M5biusgeometry；harmonicM5biUScurvaturetensor；Einsteinmna ifold MRf2000)SubjectClassification：53A03／CLCnumber：0186 Documentcode：A ArticleID： 1000．0917(2008)01．0057-10 0Introduction Let ：Mn__+Sn+beahypersurfaceinthe(几+1)．dimensionalunitsphereSn+without umbilics．Let{et)bealocalorthonormalbasisforthefirstfundamentalofrmI=dx．dxwith dualbasis )-LetII=∑巧hij8i8jbethesecondfundamentalofrmof andH=元1∑thii themeancurvatureof ．Wedefine P2 n ( j一n日) ， whereJJ_Jjisthenormwithrespecttotheinducedmetricdx·dxo