Finite Topological Type and Volume Growth

Abstract

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature RicM ≥ −(n − 1) and conjugateradius conjM ≥ c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space Hn(−1)of sectional curvature −1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.