The Compact Pseudo-Umbilical Submanifold With Parallel Mean Curvature Vector

In this paper,we study the compact pseudo-umbilical submanifold Mn with parallel mean curvature vector in Riemann manifold Nn|p.By means of the moving frame method and the Hopf maximum principle,we study the pinching problem of submanifold,namely we get some sufficient conditions for Mn to be a totally umbilical submanifold of Nn+1 by estimating the Laplacian of the square of the length of the second fundamental form and giving some restrictions to the square of the length of the second fundamental form,the Ricci curvature or the sectional curvature.In the first two chapters of this paper,we introduce the research status of submanifolds and the basic knowledge of Riemann manifolds and submanifolds,and in the last three chapters we get main conclusions and give their process of proof.The pseudo umbilical submanifolds with parallel mean curvature in constant curvature manifold are discussed in third chapter discusses,and some conclusions in the third chapter are respectively extended to locally symmetric ?-Pinching manifold and locally symmetric quasi constant curvature manifold in the last two chapters of this paper.