Submanifolds With Parallel Mean Curvature On A Sphere

In this paper, we study the Riemannian m-submanifolds in Riemannian m + p-manifolds with parallel mean curvature. In order to get the property of totally umbilical submanifolds, we introduce a function f(x). By calculatingΔf, we can get an estimate about /(x) which implies a totally umbilical submanifold.In chapter 1, we will make a general description on the recent researches in our field.In chapter 2, we discuss the pinching problems on Riemannian manifolds with parallel mean curvature and prove that Riemannian m-submanifolds with S, squareof the length of second fundamental form, less than (2m)/((m-2)^1/2+1) when m≥6 or lessthan (2m/3) when 5≥m≥2 are, in fact, to be totally umbilical.