Necessary and Sufficient Conditions for a Triangle Comparison Theorem
Differential Geometry
2018-06-13 v1
Abstract
We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.
Cite
@article{arxiv.1806.04633,
title = {Necessary and Sufficient Conditions for a Triangle Comparison Theorem},
author = {James J. Hebda and Yutaka Ikeda},
journal= {arXiv preprint arXiv:1806.04633},
year = {2018}
}