Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

Details Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

2010-04-22

arxiv.org/abs/1004.3826v3

Differential Geometry and its Applications, vol. 29 (2011) 597-605

Mathematics

Differential Geometry

This version 3 (13 pages, no figures) is a version to appear in Differential Geometry and its Applications

Scientific paper

10.1016/j.difgeo.2011.04.040

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.

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