De Sitter空间中具有平行平均曲率的类空子流形

Space - like Submanifolds with Parallel Mean Curvature in De Sitter Spaces

研究了De Sitter空间中子流形的余维数减少问题,证明了:设M是De Sitter空间Spn+p(1)中具有平行平 场曲率向量η的n维完备类空子流形,若截面曲率下确界K>0,则M是位于Spn+p(1)的全测地子流形S1n+1(1)中的 全脐超曲面,且M等距于球面 ,S<n,其中S表示第二基本形式长度的平方．

This paper studies the codimension decreasing of submanifold in a De Sitter space. It shows that let M be a complete space -like n -dimensional submanifold in the de Sitter space Sp^n＋P（ 1 ） with parallel mean curvature ,η and if K 〉0,where K is the infinimum of the sectional curvature of M, then M is totally umbilical and lies in a totally geodesic submanifold S1^n＋1 （1）of Sp^n＋p,moreover,M is isometric to a sphere S^n（√n/n-s,S〈n where S is the square of length of the second fundamental form.