Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequality

Abstract

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 ℜK 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.