S^(n+1)中Mebius形式平行的超曲面

The Hypersurfaces with Parallel Moebius Form in Sn+1

如果x:M→Sn+1是不含脐点的超曲面,且M的Moebius形式 =0和Blaschke张量A=λg,就称M为Moebius迷向超曲面,如果x:M→Sn+1 是不合脐点的超曲面,且M的Moebius形式 平行( =0)和Blaschke张量A=λg,就称M为Moebius拟迷向超曲面,这里g是M上的Moebius度量,λ:M→R是M上的光滑函数,本文证明了如下结果: (1)设x:M→Sn+1(n 3)是不含脐点的超曲面,则M是拟迷向超曲面当且仅当M是迷向超曲面,(2)设x:M→Sn+1(n 3)是不合脐点的超曲面,且M的Moebius形式 平行和Blaschke张量A也平行( A=0),则 =0.

Let x : M → Sn+1 be a hypersurface without umbilics. x is said to be Moebius isotropic if the Moebius form = 0 and the Blaschke tensor A = λg. x is said to be Moebius pseudo-isotropic if the Moebius form is parallel ( = 0) and A = λg. Here g is the Moebius metric and A : M → R is a smooth function on M. In this paper we proved the following theorem: (i) Let x : M → Sn+1(n 3) be a hypersurface without umbilics, then x is Moebius pseudo-isotropic if and only if x is Moebius isotropic. (ii) Let x : M → Sn+1(n 3) be a hypersurface without umbilics, if the Moebius form is parallel ( = 0), and A is also parallel (VA = 0), then (p = 0.