International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 3, Pages 183-191

Global pinching theorems of submanifolds in spheres

Kairen Cai

Department of Mathematics, Hangzhou Teacher's College, 96 Wen Yi Road, Hangzhou 310036, China

Received 8 June 2001

Copyright © 2002 Kairen Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let M be a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphere Sn+p(n2,p1). By using the Sobolev inequalities of P. Li (1980) to Lp estimate for the square length σ of the second fundamental form and the norm of a tensor Φ, related to the second fundamental form, we set up some rigidity theorems. Denote by σp the Lp norm of σ and H the constant mean curvature of M. It is shown that there is a constant C depending only on n, H, and k where (n1)k is the lower bound of Ricci curvature such that if σn/2<C, then M is a totally umbilic hypersurface in the sphere Sn+1.