Research Of Some Problems On Manifolds With Lower Ricci Curvature Bounds

This paper mainly consists of two parts.First,we will introduce an important generalization of classic Bonnet-Myers theorem from reference [1] in detail.This generalization extends a result from Calabi half a century ago.Second,we show that the integral of the kth(k > 0)power of distance function on an n-dimensional compact Riemannian manifold with Ricci curvature bounded below by(n-1)K is bounded below by the product of the volume of the manifold,the kth power of the diameter of the manifold and a constant which only depends on the manifold itself and power k.More specifically,when K < 0,this constant depends on K,n,k and the diameter;But when K ≥ 0,this constant is only related to n and k.We also show further that the integral of distance function on the finite ball in a noncompact Riemannian manifold with lower Ricci curvature bound has similar properties.