In this paper,the relationship between the existence of closed geodesies and the volume growth of complete noncompact Riemannian manifolds is studied.First the authors prove a diffeomorphic result of such an n-manifold with nonnegative sectional curvature,which improves Marenich-Toponogov's theorem.As an application,a rigidity theorem is obtained for nonnegatively curved open manifold which contains a closed geodesic.Next the authors prove a theorem about the nonexistence of closed geodesies for Riemannian manifolds with sectional curvature bounded from below by a negative constant.
