单位球面S^6中的Blaschke等参超曲面  

The Blaschke isoparametric hypersurfaces in the unit sphere S^6

作　　者：李兴校[1] 彭业娟[1] 

机构地区：[1]河南师范大学数学与信息科学学院,新乡453007

出　　处：《中国科学：数学》2010年第9期881-900,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:10671181);河南省基础与前沿技术计划(批准号:092300410143);河南省教育厅自然科学研究计划(批准号:2009A110010)资助项目

摘　　要：单位球面中的一个无脐点浸入子流形称为Blaschke等参子流形如果它的Mbius形式恒为零并且所有的Blaschke特征值均为常数.维数m4的Blaschke等参超曲面已经有了完全的分类.截止目前,Mbius等参超曲面的所有已知例子都是Blaschke等参的.另一方面,确实存在许多不是Mbius等参的Blaschke等参超曲面,它们都具有不超过两个的不同Blaschke特征值.在已有分类定理的基础上,本文对于5维Blaschke等参超曲面进行了完全的分类.特别地,我们证明了S6中具有多于两个不同Blaschke特征值的Blaschke等参超曲面一定是Mbius等参的,给出了此前一个问题的部分解答.An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blasehke eigenvalues are constant. The classification of Blaschke isoparametric hypersurfaces of dimension m ≤4 has already been done. Note that all known examples of MSbius isoparametric hypersurfaces are in fact Blaschke isoparametric. But there do exist many examples of Blaschke isoparametric hypersurfaces which are not Mobius isoparametric, and all those examples have no more than two Blaschke eigenvalues. In this paper we give a complete classification of Blaschke isoparametric hy- persurfaces of dimension m = 5, on the basis of known classification theorems for both the Blaschke and the MSbius isoparametric hypersurfaces. In particular, we shall prove that all the Blaschke isoparametric hypersur- faces in S6 with more than two distinct Blaschke eigenvalues are necessarily M6bius isoparametric, providing another partial solution to the problem which asks whether or not a Blasehke isoparametrie hypersurface with more than two Blasehke eigenvalues is necessarily Mobius isoparametric.

关 键 词：Blaschke等参超曲面 Mobius形式 BLASCHKE张量 Mobius度量 Mobius第二基本形式 

分 类 号：O186.1[理学—数学]