单位球面中具平行单位平均曲率向量的子流形

Submanifolds with Parallel Normalized Mean Curvature Vector in a Sphere

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^(n+p),具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^(1/2)},其中S是M的第二基本形式长度的平方,则M是S^(n+p)的一个(n+1)-维全测地子流形S^(n+1)中的超曲面。

Let M be a closed n-dimensional Riemannian manifold immersed in a unit sphere Sn+p ,p≥2 , with parallel normalized mean curvature vector. Denote by 5 the square of the length of the second fundamental form of M. It is proved that if S ≤min(2n/3, 2 ), then M is a hypersurface of a (n +1)-dimensional totally geodesic submanifold Sn+1 of Sn+p . This improve a result of Mo Xiaohuan.