On Toponogov's comparison theorem for Alexandrov spaces

TL;DR

This paper provides a clear proof of Toponogov's comparison theorem for Alexandrov spaces, extending the result to non-locally compact spaces by adapting existing arguments.

Contribution

It offers a transparent proof of Toponogov's theorem for Alexandrov spaces without assuming local compactness, broadening the theorem's applicability.

Findings

Comparison theorems hold globally under local conditions

Extension to non-locally compact Alexandrov spaces

Modification of Plaut's argument for the proof

Abstract

In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.