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

TL;DR

This paper proves that complete open manifolds with radial curvature bounded below by a von Mangoldt surface of revolution with total curvature exceeding π have finite topological type and compact isometry group, based on Busemann functions.

Contribution

It establishes that such manifolds have all Busemann functions as exhaustions and contain a compact set with all critical points, implying finite topological type.

Findings

All Busemann functions are exhaustions on the manifold.

Existence of a compact set containing all critical points.

Manifold has finite topological type and compact isometry group.

Abstract

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.