单位球面S^(n+1)中仿Blaschke特征值为常数的超曲面

The hypersurfaces with constant parablaschke eigenvalues in S^(n+1)

设M是(n+1)-维单位球面中不含脐点的超曲面,在M上可以定义所谓的Mbius度量,Mbius第二基本形式,Blaschke张量和Mbius形式,它们都是M在(n+1)-维单位球面中的Mbius变换群下的不变量.对称的(0,2)张量D=A+λB也是Mbius不变量,称为浸入x的仿Blaschke张量,其中λ是常数,仿Blaschke张量的特征值称为仿Blaschke特征值.本文对满足条件(1)Φ=0;(2)D平行且具有三个互异的常特征值的超曲面进行了分类.

Let M be an immersed umbilic - free hypersurface in the （ n ＋ 1 ） - dimensional unite sphere, then M is associated with a so - called Mobius metric, a MSbius second fundamental form, a Blaschke tensor and a Mobius form which are invariants of M under the Mobius transformation group of （ n ＋ 1 ） - dimensional unite sphere. In this paper,letting D =A ＋AB,where A is a constant,then D is a symmetric （0,2） tensor and a Mobius invariant. D is called Parablaschke tensor of x, an eigenvalue of the Parablaschke tensor is called a Parablaschke eigenvalue of x. We classify the hypersurfaces x ： Mn→Sn＋1 , which satisfy ： （ 1 ） Ф = 0 ; （ 2 ） D is parallel and has exact three distinct Parablaschke eigenvalues.