Personal tools

Replacing the Lower Curvature Bound in Toponogov's Comparison Theorem by a Weaker Condition

James Hebda, SLU

  • Geometry/ Topology Seminar
  • Topology Seminar
When Tue, Dec 01, 2015
from 02:10 PM to 03:00 PM
Where 225 Ritter Hall
Contact Name
Add event to calendar vCal
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 some 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 talk, I will discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than 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.

(This is joint work with Yutaka Ikeda.)
« March 2018 »
Upcoming Events
Math/CS Club
Wed, Mar 21, 2018
100 prisoners Bryan Clair, SLU
Fri, Mar 23, 2018
Introduction to Cluster Algebras Kyungyong Lee, University of Nebraska, Lincoln
Wed, Apr 18, 2018
Quantitative Topology? by Shmuel Weinberger, U Chicago
Annual Awards Ceremony
Fri, Apr 27, 2018
The 2018 Math & Statistics Department Award Ceremony Featuring Sarah Greenwald, Appalachian State University
Previous events…
Upcoming events…