We consider the previously introduced notion of the -quadrilateral cosine, which is the cosine under parallel transport in model -space, and which is denoted by . In -space, 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 if ) is an domain (otherwise known as a space) if and only if always or always . (We prove that in such spaces always is equivalent to always .) The case of was treated in our previous paper on quasilinearization. We show that in our theorem the diameter hypothesis for positive is sharp, and we prove an extremal theorem—isometry with a section of -plane—when attains an upper bound of , the case of equality in the metric Cauchy–Schwarz inequality. We derive from our main theorem and our previous result for a complete solution of Gromov’s curvature problem in the context of Aleksandrov spaces of curvature bounded above.
"Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequality." Michigan Math. J. 67 (2) 289 - 332, May 2018. https://doi.org/10.1307/mmj/1519095621