The Rigid Problems Of Submanifolds In Riemannian Manifolds

In the paper, we study the Submanifolds in Riemannian Manifolds. Specifically speaking,there topics are discussed.The first is the preliminaries. The second is the compact minimalSubmanifolds in locally symmetric space; the third is the compact pseudo-umbilical submanifoldsin locally symmetric manifolds.In Chapter 1. We give some basic knowledge and lemma of the n+p dimensional RiemannianManifold with sectional curvature KN satisfies 21 <δKN 1.In Chapter 2. We study the compact minimal Submanifolds of n+p dimension in locallysymmetric space.Let Nn+p be a n+p-dimensional locally symmetric complete Riemannian Man-ifold with sectional curvature KN satisfies 21 <δKN 1 and Mn be an n-dimensionalcompact minimal Submanifolds in Nn+p .In this paper,the authors discuss the pinching theoremabout this manifolds with Ricci curvature and the sectional curvature.In Chapter 3. We study the compact pseudo-umbilical Submanifolds of n+p dimensionin locally symmetric manifolds.Let Nn+p be a n+p-dimensional locally symmetric completeRiemannian Manifold with sectional curvature KN satisfies 21 <δKN 1 and Mn be an-dimensional compact pseudo-umbilical Submanifolds in Nn+p .we receive this kind of Sub-manifolds of intrinsic rigidity theorem.