TY - JOUR

T1 - Characterization of aleksandrov spaces of curvature bounded above by means of the metric cauchy–schwarz inequality

AU - Berg, I. D.

AU - Nikolaev, I. G.

PY - 2018/5

Y1 - 2018/5

N2 - We consider the previously introduced notion of the Kquadrilateral 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 p/(2K) if K > 0) is an K domain (otherwise known as a CAT(K) space) if and only if always cosqK = 1 or always cosqK = -1. (We prove that in such spaces always cosqK = 1 is equivalent to always cosqK = -1.) 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.

AB - We consider the previously introduced notion of the Kquadrilateral 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 p/(2K) if K > 0) is an K domain (otherwise known as a CAT(K) space) if and only if always cosqK = 1 or always cosqK = -1. (We prove that in such spaces always cosqK = 1 is equivalent to always cosqK = -1.) 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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U2 - 10.1307/mmj/1519095621

DO - 10.1307/mmj/1519095621

M3 - Article

AN - SCOPUS:85047179165

VL - 67

SP - 289

EP - 332

JO - Michigan Mathematical Journal

JF - Michigan Mathematical Journal

SN - 0026-2285

IS - 2

ER -