球面的正曲率子流形

SUBMANIFLOD WITH POSITIVE CURVATURE IN A SPHERE

研究正常曲率流形的子流形的余维数减少问题,证明:若n+p维正常曲率c的黎曼流形的n维紧致子流形M有l维法子从N1,使得平均曲率向量平行和位于N1中且N1存在平行的幺正标架以及k>0,S-nH2>n(p-l)(c-2K),其中K是截面曲率下确界,S是第二基本形式长度平方,H是平均曲率,则M是N的n+l维全测地子流形中的全脐超曲面,从而是常曲率的。改进了徐森林等[3]中的定理。

This paper studies the codimension decreasing of submanifold in a constant curved manifold.It shows that let M be an n-dimensional compact submanifold of the n+p-dimensional Riemannian manifold N with the positire constant curvature c and if there is a normal sublundle N_1 of fiber dimension l such that the mean curvature vector is parallel and lies in N_1, and there is a parallel orthonarmal frame in N_1, and K>0,S-nH^2>n(p-l)(c-2K),where K is the infimum of the section curvature of M,S the square of length of the tecond fun damental form,and H mean curvature ,then M is a totally umbilical hypersurface in n+1 dimensional totally geodesic submanifold of N,and has constant sectional curvature.