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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 $\operatorname{cosq}_{K}$ . In $K$ -space, $\vert \operatorname{cosq}_{K}\vert \leq 1$ 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 $\pi /(2\sqrt{K})$ if $K\gt 0$ ) is an $\Re_{K}$ domain (otherwise known as a $\operatorname{CAT}(K)$ space) if and only if always $\operatorname{cosq}_{K}\leq 1$ or always $\operatorname{cosq}_{K}\geq -1$ . (We prove that in such spaces always $\operatorname{cosq}_{K}\leq 1$ is equivalent to always $\operatorname{cosq}_{K}\geq -1$ .) 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 $\vert \operatorname{cosq}_{K}\vert $ 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.

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Autor(es)

Berg, I. D. -  Nikolaev, Igor G. - 

Id.: 71305349

Idioma: inglés  - 

Versión: 1.0

Estado: Final

Tipo:  application/pdf - 

Palabras clave53C20 - 

Tipo de recurso: Text  - 

Tipo de Interactividad: Expositivo

Nivel de Interactividad: muy bajo

Audiencia: Estudiante  -  Profesor  -  Autor  - 

Estructura: Atomic

Coste: no

Copyright: sí

: Copyright 2018 The University of Michigan

Formatos:  application/pdf - 

Requerimientos técnicos:  Browser: Any - 

Relación: [References] 0026-2285
[References] 1945-2365

Fecha de contribución: 13-may-2018

Contacto:

Localización:
* Michigan Math. J. 67, iss. 2 (2018), 289-332
* doi:10.1307/mmj/1519095621

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