International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 3, Pages 183-191
Global pinching theorems of submanifolds in spheres
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 be a compact embedded submanifold with parallel mean
curvature vector and positive Ricci curvature in the unit sphere
. By using the Sobolev
inequalities of P. Li (1980) to estimate for the square length
of the second fundamental form and the norm of a tensor
, related to the second fundamental form, we set up some
rigidity theorems. Denote by the norm of
and the constant mean curvature of . It is shown that there is a constant depending only on , , and where is the lower bound of Ricci curvature such that if
, then is a totally umbilic hypersurface in the sphere .