On the Asymptotic Geometry of Nonpositively Curved Manifolds

In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces. Submitted: August 1996, final version: April 1997