摘要
Let M<sup>n</sup> be an n-dimensional compact submanifold in a unit sphere S<sup>n+p</sup>, S be the square of the length of the second fundamental form of M<sup>n</sup>. S. T. Yau proved that if the mean curvature vector is parallel and S≤n/(n<sup>1/2</sup>+3-1/（p-1）) everywhere on M<sup>n</sup> then M<sup>n</sup> lies in a totally geodesic S<sup>n+1</sup>. The constant has been improved to max,
基金
Project supported partially by the National Natural Science Foundation of China
