S~5上仿Blaschke张量的特征值为常数的超曲面

设x:M→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,其中λ是常数,D称为浸入x的仿Blaschke张量.李海中和王长平研究了满足条件:(i)Φ=0;(ii)A+λB+μg=0的超曲面,其中λ和μ都是函数,他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是在Φ=0的条件下D只有一个互异的特征值的超曲面的分类.本文对S~5上满足如下条件的超曲面进行了完全分类:(i)Φ=0,(ii)对某常数λ,D具有常数特征值.

The Hypersurfaces in S~5 with Constant Para-Blaschke Eigenvalues

Let x:M→Sⁿ⁺¹ be a hypersurface in the (n+1)-dimensional unit sphere Sⁿ⁺¹ without umbilics. Four basic invariants of x under the Moebius transformation group in Sⁿ⁺¹ are a Riemannian metric g called Moebius metric, a 1-form Φ called Moebius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Moebius second fundamental form. Let D =A+λB, where λ is a constant. Then D is a symmetric (0,2) tensor and a Moebius invariant. D is called Para-Blaschke tensor of x. Li and Wang have studied the hypersurfaces x:M→Sⁿ⁺¹, which satisfy: (i) Φ=0, (ii) A+λB+μg=0 for some functions λ and μ on M; they have proved that λ and μ must be constants and have classified the hypersurfaces; in fact, they have classified the hypersurfaces which satisfy: (i) Φ=0, (ii) D has only one distinct constant eigenvalue. In this paper, We classify the hypersurfaces x:M→S⁵, which satisfy: (i) Φ=0, (ii) D has constant eigenvalues for some constants λ.