A generalized maximal diameter sphere theorem
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 excee...
Gespeichert in:
1. Verfasser: | |
---|---|
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | 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. |
---|---|
DOI: | 10.48550/arxiv.1607.05011 |