On Toponogov's comparison theorem for Alexandrov spaces
Urs Lang, Viktor Schroeder

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology
