Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature

TL;DR

This paper investigates the topology of noncompact manifolds with asymptotically nonnegative Ricci curvature, showing they are often diffeomorphic to Euclidean space under certain curvature decay and volume growth conditions.

Contribution

It establishes new conditions under which manifolds with asymptotically nonnegative Ricci curvature are diffeomorphic to Euclidean space, extending previous topological classification results.

Findings

Manifolds with asymptotically nonnegative Ricci curvature can be diffeomorphic to R^n.

Quadratic decay of sectional curvature influences topological structure.

Volume growth conditions affect the diffeomorphism to Euclidean space.

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 with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space 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.