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

TL;DR

This paper generalizes the characterization of Aleksandrov spaces with curvature bounds using the $K$-quadrilateral cosine, providing necessary and sufficient conditions for such spaces to be $ ext{CAT}(K)$ spaces, extending previous results for zero curvature.

Contribution

It introduces the $K$-quadrilateral cosine as a key tool to characterize $ ext{CAT}(K)$ spaces for arbitrary curvature bounds, extending previous zero-curvature results.

Findings

Characterization of $ ext{CAT}(K)$ spaces via $ ext{cosq}_K$ bounds.

Necessary and sufficient conditions for complete $ ext{Re}_K$ domains.

Sharpness of the diameter hypothesis for positive $K$.

Abstract

In this note, we extend the main results of our paper on quasilinearization and curvature of Aleksandrov spaces of curvature $\leq0$ 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 $\operatorname{cosq}_{K}$. 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 $\operatorname{CAT}\left(K\right) $ space) if and only if always $\operatorname{cosq}_{K}\leq1$ or always $\operatorname{cosq}_{K}$ $\geq-1$. (We prove that in such spaces always $\operatorname{cosq}_{K}\leq1$ is equivalent to always $\operatorname{cosq}% _{K}$ $\geq-1$). 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 $|\operatorname{cosq}_{K}|$ 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.