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

Abstract

This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold $M$. In the first series [KT1], we showed that all Busemann functions on an $M$ which is not less curved than a von Mangoldt surface of revolution $\widetilde{M}$ are exhaustions, if the total curvature of $\widetilde{M}$ is greater than π. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to $\boldsymbol{R}^{2}$ whose Gaussian curvature is non-increasing along each meridian. Our purpose of this series is to generalize the main theorem in [KT1] to an $M$ which is not less curved than a more general surface of revolution.