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 is homeomorphic to the interior of a compact manifold with boundary if the manifold is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than . 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.
"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." Nagoya Math. J. 209 23 - 34, March 2013. https://doi.org/10.1215/00277630-1959451