The Gauss Curvature of a model surface with finite total curvature is not always bounded

Details The Gauss Curvature of a model surface with finite total curvature is not always bounded The Gauss Curvature of a model surface with finite total curvature is not always bounded

2011-02-04

arxiv.org/abs/1102.0852v2

Mathematics

Differential Geometry

A few minor corrections had been done in the version 2. 10 pages, no figures

Scientific paper

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.

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