The Volume Growth Of Complete Noncompact Riemannian Manifold With Nonnegative Curvature

In this thesis, we mainly study the volume growth of complete noncompact Riemannian manifold with nonnegative curvature, which has relations with closed geodesics and critical points of distance functions.To be precise, we proved the following two results:Theorem1 Let M~n be a complete noncompact manifold with nonnegative curvature. If M~n contains a nontrivial closed geodesic (i. e. not apoint), thenα_M =0. On the other words, ifα_M >0, then M~n does not contain anynontrivial closed geodesic.Theorem2 Let M~n be a complete noncompact manifold with nonnegativecurvature. Ifα_M > 1/2, then any distance function d(p,x) has no critical point except p .