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In this note, we extend the main results of our paper on quasilinearizationand curvature of Aleksandrov spaces of curvature $\leq0$ to curvature boundsother than $0$. For non-zero $K$, we employ the previously introduced notion ofthe $K$-quadrilateral cosine, which is the cosine under parallel transport inmodel $K$-space, and which is denoted by $\operatorname{cosq}_{K}$. Ourprincipal result states that a geodesically connected metric space (of diameternot 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 ifalways $\operatorname{cosq}_{K}\leq1$ 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 semimetricspace to be a complete $\Re_{K}$ domain. We show that in our theorem thediameter hypothesis for positive $K$ is sharp and we prove an extremal theoremwhen $|\operatorname{cosq}_{K}|$ attains an upper bound of $1$. We derive fromour main theorem and our previous result for $K=0$ a complete solution ofGromov's curvature problem in the context of Aleksandrov spaces of curvaturebounded above. Then we establish the $K$-Euler's inequality and the extremaltheorem for equality in the $K$-Euler's inequality in an $\Re_{K}$ domain.

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