Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature

Abstract

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold $M$ with asymptotically nonnegative Ricci curvature and sectional curvature $K(x)\ge \frac{C}{d_{p}(x)^{\alpha}}$ is diffeomorphic to the Euclidean space $\mathbb{R}^n$ under some conditions on the density of rays starting from the base point $p$ or on the volume growth of geodesic balls in $M$.