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On Toponogov's comparison theorem for Alexandrov spaces


Lang, Urs; Schroeder, Viktor (2013). On Toponogov's comparison theorem for Alexandrov spaces. L'Enseignement Mathématique, 59(3-4):325-336.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2013
Deposited On:03 Oct 2014 12:09
Last Modified:05 Apr 2016 18:23
Publisher:European Mathematical Society
ISSN:0013-8584 (P) 2309-4672 (E)
Publisher DOI:https://doi.org/10.4171/LEM/59-3-6
Related URLs:http://www.ams.org/mathscinet-getitem?mr=3189039 (Organisation)

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