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

TL;DR

This paper investigates how the total curvature of model surfaces influences the topology of complete open Riemannian manifolds with radial curvature bounds, extending previous results to more general surfaces of revolution.

Contribution

It generalizes earlier theorems by showing that Busemann functions are exhaustions under broader curvature conditions involving general surfaces of revolution.

Findings

Busemann functions are exhaustions when total curvature exceeds π.

Extension of previous results to more general surfaces of revolution.

Provides topological control of manifolds via curvature bounds.

Abstract

Dedicated to Professor K. Shiohama on the occasion of his seventieth birthday: This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold $M$. In the first series [arXiv:0901.4010], we showed that all Busemann functions on an $M$ which is not less curved than a von Mangoldt surface of revolution are exhaustions, if the total curvature of the surface is greater than $\pi$. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to Euclidean plane whose Gaussian curvature is non-increasing along each meridian. Our purpose of this series is to generalize the main theorem in [arXiv:0901.4010] to an $M$ which is not less curved than a more general surface of revolution.