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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