Rigidity of compact submanifolds in a unit sphere

Abstract

In this paper, we prove a Pinching theorem for compact submanifolds with non-zero parallel mean curvature, which improve the Pinching constant in [5]. For lower dimensional compact submanifolds we obtain a strong result. Meanwhile, we study the Pinching problem for the sectional curvatures of minimals submanifolds, and obtain the best Pinching constant so far.