Tohoku Mathematical Journal
2019
March
SECOND SERIES VOL. 71, NO. 1

Tohoku Math. J.
71 (2019), 145-155

Title A GENERALIZED MAXIMAL DIAMETER SPHERE THEOREM

Author Nathaphon Boonnam

(Received July 25, 2016, revised October 11, 2016)
Abstract. We prove that 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 $\widetilde M$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\widetilde M$. Moreover, we prove that if the diameter of $M$ equals that of $\widetilde M$, then $M$ is isometric to the $n$-model of $\widetilde M$. The class of a two-sphere of revolution employed in our main theorem is very wide. For example, this class contains both ellipsoids of prolate type and spheres of constant sectional curvature. Thus our theorem contains both the maximal diameter sphere theorem proved by Toponogov [9] and the radial curvature version by the present author [2] as a corollary.

Mathematics Subject Classification. Primary 53C22.
Key words and phrases. Cut locus, generalized first variation formula, geodesic triangle, maximal diameter sphere theorem, Toponogov comparison theorem, two-sphere of revolution, radial sectional curvature.

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