A generalized maximal diameter sphere theorem (Q2418806)

The main result of the paper is the following: If a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\tilde{M}$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\tilde{M}$. Moreover, if the diameter of $M$ equals that of $\tilde{M}$, then $M$ is isometric to the $n$-model of $\tilde{M}$. The class of a two-sphere of revolution employed in this theorem contains both ellipsoids of prolate type and spheres of constant sectional curvature. This theorem generalizes a maximal diameter sphere theorem proved by \textit{V. A. Toponogov} [Usp. Mat. Nauk 14, No. 1(85), 87--130 (1959; Zbl 0114.37504)] and a radial curvature version proved by the author in [Tokyo J. Math. 38, No. 1, 145--151 (2015; Zbl 1327.53048)].