A generalized maximal diameter sphere theorem
| Autor: | Nathaphon Boonnam |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics generalized first variation formula geodesic triangle General Mathematics Toponogov comparison theorem Mathematical analysis Cut locus maximal diameter sphere theorem Riemannian manifold Type (model theory) Curvature 53C22 radial sectional curvature Base (group theory) cut locus Differential Geometry (math.DG) Bounded function FOS: Mathematics two-sphere of revolution Sphere theorem Sectional curvature Mathematics::Differential Geometry Mathematics |
| Zdroj: | Tohoku Math. J. (2) 71, no. 1 (2019), 145-155 |
| Popis: | 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. 11 pages, no figure |
| Databáze: | OpenAIRE |
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