Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. I

Details Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. I Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. I

2009-01-26

arxiv.org/abs/0901.4010v2

Mathematische Annalen, Vol. 351, Issue 2 (2011), Page 251-266

Mathematics

Differential Geometry

This version 2 is a version to appear in Math. Ann. 16 pages, No figures

Scientific paper

10.1007/s00208-010-0593-4

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than $\pi$. We show, as our main theorem, that all Busemann functions on $M$ are exhaustions, and that there exists a compact subset of $M$ such that the compact set contains all critical points for any Busemann function on $M$. As corollaries by the main theorem, $M$ has finite topological type, and the isometry group of $M$ is compact.

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