Pinching Problems Of Sub Manifolds

By the application of The technique of J.Simons,there have been many rigidity results obtained for minimal sub manifolds or for sub manifolds with parallel mean curvature vector field immersed into a unit sphere.The thesis mainly to study the pinching problems of submanifolds,we prove rigidity theorems of several submanifolds.In the first section,we will make a general description on the recent researches in our field.In the second section,we discuss the most fundamental theory of Riemann geometry.In the third section,Let Mⁿ be a submanifold with parallel mean Curvature in a locally symmetric Riemannian manifold.We prove a integral about the square of the second fundamental form of M and its pinching theorem.In the forth section,we discuss the oriented closed pseudo-umbilical submanifold Mⁿ with parallel mean curvature vector in a Riemannian manifold N^n+p.We prove a integral about the square of the second fundamental form of M.In fifth section,let Mⁿ be a minimal submanifold with parallel Ricci Curvature Vector in Riemannian manifold N^n+p.We prove a integral about the square of the second fundamental form of M and its pinching theorem.