Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

TL;DR

This paper establishes conditions under which complete non-compact Riemannian manifolds are diffeomorphic to Euclidean spaces, focusing on radial Ricci curvature bounds and volume growth without relying on Gromov-Hausdorff convergence.

Contribution

It provides new sufficient conditions for manifolds to be diffeomorphic to Euclidean spaces based on radial Ricci curvature and volume growth, avoiding Gromov-Hausdorff techniques.

Findings

Radial Ricci curvature bounded below by a model curvature function.

Manifold diffeomorphic to Euclidean space if model volume growth is close to 1.

Generalizes results of do Carmo and Changyu.

Abstract

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case.