A $K$-quadrilateral cosine characterization of Aleksandrov spaces of curvature bounded above

Abstract

In this note, we extend the main results of our paper on quasilinearization and curvature of Aleksandrov spaces of curvature $\le 0$ to curvature bounds other than $0$. For non-zero $K$, we employ 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 ${\mathrm{cosq}}_K$. 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 $\mathrm{CAT}\left(K\right)$ space) if and only if always ${\mathrm{cosq}}_K\le 1$ or always ${\mathrm{cosq}}_K$ $\ge -1$. (We prove that in such spaces always ${\mathrm{cosq}}_K\le 1$ is equivalent to always $\mathrm{cosq}$ $\ge -1$). As a corollary, we give necessary and sufficient conditions for a Cauchy complete semimetric space to be a complete ${\Re }_K$ domain. We show that in our theorem the diameter hypothesis for positive $K$ is sharp and we prove an extremal theorem when $|{\mathrm{cosq}}_K|$ attains an upper bound of $1$. 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. Then we establish the $K$-Euler's inequality and the extremal theorem for equality in the $K$-Euler's inequality in an ${\Re }_K$ domain.