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

摘要

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 cosq_K. 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 π/(2K~(1/2)) if K > 0) is an R_K domain (otherwise known as aCAT(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.