A proof of Toponogov's theorem in Alexandrov geometry

TL;DR

This paper provides an elementary proof of Toponogov's theorem within Alexandrov geometry, utilizing concepts inspired by Riemannian sectional curvature and second variation formulas.

Contribution

It introduces a simplified proof of Toponogov's theorem in Alexandrov spaces, bridging ideas from Riemannian geometry.

Findings

Elementary proof of Toponogov's theorem established

Connection between Alexandrov and Riemannian curvature concepts clarified

Potential for broader applications in metric geometry

Abstract

This paper aims to give an elementary proof for Toponogov's theorem in Alexandrov geometry with lower curvature bound. The idea of the proof comes from the fact that, in Riemannian geometry, sectional curvature can be embodied in the second variation formula.