The Submanifolds With Constant Mean Curvature In A Manifold Of Quasi-Constant Curvature

The submanifolds theory is an important part of the fundamental research in differential geometry. In this paper, getting inspiration from the classic Simons in-equality in constant curvature space, we extended the outer space and submanifolds respectively to quasi-constant curvature Riemannian manifold and it’s submanifolds with constant mean curvature. We obtain some integral inequality corresponding to above submanifolds by estimating Laplacian of the square of the length of the second fundamental form, by limiting the submanifolds and outer space we get some basic conclusions.This article is divided into three chapters altogether. The first chapter intro-duces the investigation status of submanifolds the theory, to understand the main research content and method, and explain the intention of this article. The second chapter presents some basic concepts and theorems in differential geometry. The third chapter is the main part of this article, giving some basic computing and lem-ma, then proving the main conclusions of this paper.