On Toponogov's comparison theorem for Alexandrov spaces
Urs Lang, Viktor Schroeder
(Submitted on 16 Jul 2012 (v1), last revised 25 Jul 2012 (this version, v2))
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.
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