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We will construct surfaces of revolution with finite total curvature whoseGauss curvatures are not bounded. Such a surface of revolution is employed as areference surface of comparison theorems in radial curvature geometry.Moreover, we will prove that a complete non-compact Riemannian manifold M ishomeomorphic to the interior of a compact manifold with boundary, if themanifold 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 ofmetrics than that of a complete non-compact Riemannian manifold whose sectionalcurvature is 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