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

Abstract

We construct distinctive 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 prove that a complete noncompact Riemannian manifold $M$ is homeomorphic to the interior of a compact manifold with boundary if the manifold $M$ is not less curved than a noncompact model surface $\mathrm{M̃}$ of revolution and if the total curvature of the model surface $\mathrm{M̃}$ is finite and less than $2\pi$. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.