Necessary and Sufficient Conditions for a Triangle Comparison Theorem

TL;DR

This paper establishes necessary and sufficient conditions for triangle comparison theorems in Riemannian geometry using surfaces of revolution as models, extending classical results and deriving new geometric theorems.

Contribution

It provides a comprehensive set of conditions for triangle comparison theorems with model surfaces of revolution, including new versions of the Maximal Radius and Sphere Theorems.

Findings

Necessary and sufficient conditions for triangle comparison with revolution surfaces.

Extension of Topogonov's theorem to new model spaces.

Proof of Maximal Radius and Sphere Theorems under these conditions.

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.