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 ${cosq}_K$. In $K$-space, $\left|{cosq}_K\right|\le 1$ 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 $\pi /\left(2\sqrt{K}\right)$ if $K>0$) is an ${\Re }_K$ domain (otherwise known as a $CAT\left(K\right)$ space) if and only if always ${cosq}_K\le 1$ or always ${cosq}_K\ge -1$. (We prove that in such spaces always ${cosq}_K\le 1$ is equivalent to always ${cosq}_K\ge -1$.) 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 $\left|{cosq}_K\right|$ 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.