Pinching Theorems Of Submanifolds

Differential Geometry is an old subject.lt has long history. Since E. Cartan found Moving Frame, which was developed by Chen Xingshen, one branch of it, Submanifold Geometry, has been a hot spot. The Moving Frame is used to calculate: the Laplacian of different Functions.Many results of the Submanifolds in one space have been obtained. Are there similar results of two or more than two spaces?In this paper, the Submanifolds in two Nested spaces are discussed. The paper is divided into two chapters. The contents are following:In the first chapter, two inequlities for Compact Pseudo-umbilical Submanifolds with Parallel Mean Curvature Vector in Sn+p(c) are presented, this generalizing the results which have existed. For this kind of Submanifolds, two inequlities about RH are obtained, and this Submanifold is Totally Umbilical.In the second chapter ,Submanifolds in two Nested spaces are discussed. Pseudoumblical Submanifold with Parallel Mean Curvature Vector in Constant space, Quasi-constant Curvature space and Locally Symmetric space are discussed.