Total Curvatures of Model Surfaces Control Topology of Complete Open Manifolds with Radial Curvature Bounded Below. I

We investigate the finiteness structure of a complete open Riemannian n-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 π. 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.