The Alexandrov-Toponogov comparison theorem for radial curvature

Abstract

We discuss the Alexandrov-Toponogov comparison theorem under the conditions of radial curvature of a pointed manifold $(M,o)$ with reference surface of revolution $(\widetilde M, \tilde o)$. There are two obstructions to make the comparison theorem for a triangle one of whose vertices is a base point $o$. One is the cut points of another vertex $\tilde p \not=\tilde o$ of a comparison triangle in $\widetilde M$. The other is the cut points of the base point $o$ in $M$. We find a condition under which the comparison theorem is valid for any geodesic triangle with a vertex at $o$ in $M$.