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May 2018 Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequality
I. D. Berg, Igor G. Nikolaev
Michigan Math. J. 67(2): 289-332 (May 2018). DOI: 10.1307/mmj/1519095621


We consider the previously introduced notion of the K-quadrilateral cosine, which is the cosine under parallel transport in model K-space, and which is denoted by cosqK. In K-space, |cosqK|1 is equivalent to the Cauchy–Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesically connected metric space (of diameter not greater than π/(2K) if K>0) is an K domain (otherwise known as a CAT(K) space) if and only if always cosqK1 or always cosqK1. (We prove that in such spaces always cosqK1 is equivalent to always cosqK1.) The case of K=0 was treated in our previous paper on quasilinearization. We show that in our theorem the diameter hypothesis for positive K is sharp, and we prove an extremal theorem—isometry with a section of K-plane—when |cosqK| attains an upper bound of 1, the case of equality in the metric Cauchy–Schwarz inequality. We derive from our main theorem and our previous result for K=0 a complete solution of Gromov’s curvature problem in the context of Aleksandrov spaces of curvature bounded above.


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I. D. Berg. Igor G. Nikolaev. "Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequality." Michigan Math. J. 67 (2) 289 - 332, May 2018.


Received: 19 September 2016; Revised: 20 November 2017; Published: May 2018
First available in Project Euclid: 20 February 2018

zbMATH: 06914765
MathSciNet: MR3802256
Digital Object Identifier: 10.1307/mmj/1519095621

Primary: 53C20
Secondary: 51K10, 53C45

Rights: Copyright © 2018 The University of Michigan


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Vol.67 • No. 2 • May 2018
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