Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics 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 geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.