Zbl 1144.53045
Berg, I.D.; Nikolaev, I.G.
Quasilinearization and curvature of Aleksandrov spaces. (English)
[J] Geom. Dedicata 133, 195-218 (2008). ISSN 0046-5755; ISSN 1572-9168
Authors' abstract: We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine-a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space ${\left(\mathcal{M},\rho\right)}$ is an Aleksandrov ${\Re_{0}}$ domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in ${\mathcal{M}}$. We also observe that a geodesically connected metric space ${\left(\mathcal{M},\rho\right)}$ is an ${\Re_{0}}$ domain if and only if, for every quadruple of points in ${\mathcal{M}}$, the quadrilateral inequality (known as Euler's inequality in ${\mathbb{R}^{2}}$) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an ${\Re_{0}}$ domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.
[Themistocles M. Rassias (Athens)]- MSC 2000:
- *53C20 Riemannian manifolds (global)
53C45 Global surface theory (a la A.D. Aleksandrov)
51K10 Synthetic differential geometry
Keywords: Aleksandrov space; quadrilateral cosine; quadrilateral inequality; 2-roundness; curvature problem; quasi-inner product; Aleksandrov lower angle
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