The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

TL;DR

This paper constructs specific revolution surfaces with finite total curvature and unbounded Gauss curvature, and uses them to analyze the topology of non-compact manifolds with radial curvature bounds, extending classical comparison theorems.

Contribution

It introduces new model surfaces with finite total curvature and unbounded curvature, and applies them to topological classification of manifolds under radial curvature bounds.

Findings

Constructed revolution surfaces with finite total curvature and unbounded Gauss curvature.

Proved topological homeomorphism of certain manifolds to interior of compact manifolds.

Extended comparison theorems to wider classes of metrics with finite total curvature.

Abstract

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary, if the manifold M is not less curved than a non-compact model surface of revolution, and if the total curvature of the model surface is finite and less than $2\pi$. Hence, in the first result mentioned above, we may treat a much wider class of metrics than that of a complete non-compact Riemannian manifold whose sectional curvature is bounded from below by a constant.