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

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, | cosq_K | = 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 cosq_K = 1 or always cosq_K = -1. (We prove that in such spaces always cosq_K = 1 is equivalent to always cosq_K = -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 | cosq_K | 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.