On Pseudo-umbilical Submanifolds In A Quasi Constant Curvature Space

This paper mainly studies the compact pseudo umbilical submanifolds with parallel mean curvature vector in the quasi constant curvature space.We estimate the laplacian of the length of the second fundamental form by the J.Simon's method for minimal submanifold in the constant curvature space,and obtain some pinching theorems relating to the second fundamental form,sectional curvature and ricci curvature.This paper is divided into 4 chapters.The first chapter introduces research status on submanifolds in the constant curvature and quasi constant curvature space.The second chapter introduces the basic knowledge of the submanifolds.The third chapter mainly introduces the Riemann manifolds of the quasi constant curvature,and gives the basic equations of submanifold of the quasi constant curvature space.The fourth chapter is the central part of this paper,mainly studes the compact pseudo umbilical submanifolds with parallel mean curvature vector in the quasi constant curvature space,gives the related lemma and calculation,and obtaines main results in this paper which generalize relevant conclusions of constant curvature space.