A Schur-Toponogov theorem in Riemannian geometry & a new proof of Toponogov's theorem in Alexandrov geometry

TL;DR

This paper introduces a Schur-Toponogov theorem in Riemannian geometry that generalizes existing theorems and provides a new proof of Toponogov's theorem in Alexandrov geometry, linking these geometric concepts.

Contribution

It presents a unified Schur-Toponogov theorem in Riemannian geometry and offers a novel proof of Toponogov's theorem in Alexandrov geometry, highlighting their relationship.

Findings

Generalizes Schur's and Toponogov's theorems

Provides a new proof of Toponogov's theorem in Alexandrov geometry

Establishes a connection between Riemannian and Alexandrov geometric results

Abstract

In the paper, we give a Schur-Toponogov theorem in Riemannian geometry, which not only generalizes Schur's and Toponogov's theorem but also indicates their relation. Inspired by its proof, we also supply a new proof of Toponogov's theorem (in the large) in Alexandrov geometry.