The Structure Theorem for the Cut Locus of a Certain Class of Cylinders of Revolution I

Abstract

We generalize Toponogov's maximal diameter sphere theorem from the radial curvature geometry's standpoint. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of prolate type, the diameter of $M$ does not exceed the diameter of the ellipsoid. Furthermore if the diameter of such an $M$ equals that of the ellipsoid, then $M$ is isometric to the $n$-dimensional ellipsoid of revolution.