Translator Disclaimer
2017 Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis
James J. Hebda, Yutaka Ikeda
Tohoku Math. J. (2) 69(2): 305-325 (2017). DOI: 10.2748/tmj/1498269628

Abstract

Toponogov's triangle comparison theorem and its generalizations are important tools for studying the topology of Riemannian manifolds. In these theorems, one assumes that the curvature of a given manifold is bounded from below by the curvature of a model surface. The models are either of constant curvature, or, in the generalizations, rotationally symmetric about some point. One concludes that geodesic triangles in the manifold correspond to geodesic triangles in the model surface which have the same corresponding side lengths, but smaller corresponding angles. In addition, a certain rigidity holds: Whenever there is equality in one of the corresponding angles, the geodesic triangle in the surface embeds totally geodesically and isometrically in the manifold.

In this paper, we discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in the model with the same corresponding side lengths, but smaller corresponding angles. In contrast, it is interesting that rigidity fails in this setting.

Citation

Download Citation

James J. Hebda. Yutaka Ikeda. "Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis." Tohoku Math. J. (2) 69 (2) 305 - 325, 2017. https://doi.org/10.2748/tmj/1498269628

Information

Published: 2017
First available in Project Euclid: 24 June 2017

zbMATH: 1376.53061
MathSciNet: MR3682168
Digital Object Identifier: 10.2748/tmj/1498269628

Subjects:
Primary: 53C20
Secondary: 53C22

Rights: Copyright © 2017 Tohoku University

JOURNAL ARTICLE
21 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.69 • No. 2 • 2017
Back to Top