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

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.