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

中文关键词:  球面  平均曲率  单位平均曲率向量

英文关键词:sphere  mean curvature  normalized mean curvature vector.

中文摘要:

      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Sⁿ⁺¹中的超曲面。

英文摘要:

      Let M be a closed n-dimensional Riemannian manifold immersed in a unit sphere S^n+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(n-1)^1/2}, then M is a hypersurface of a (n +1)-dimensional totally geodesic submanifold Sⁿ⁺¹ of S^n+p. This improve a result of Mo Xiaohuan.