Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds
type: Item
instance of: scholarly article
title: Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds
zbMATH Open document ID: 0879.53031
published in: Michigan Mathematical Journal
publication date: 1997-12-17T00:00:00Z
review text: The following definition is due to A. D. Alexandrov. Let \(M\) be a metric space and consider a geodesic triangle \(T\) in \(M\), that is, the union of three geodesic segments (the edges of \(T\)) joining three points in \(M\) (the vertices of \(T\)). Let \(f\) be a map which sends the union of the edges of \(T\) onto the edges of a triangle in the Euclidean plane. Then \(M\) is said to satisfy the CAT(0) inequality if every such map \(f\), for every triangle \(T\) in \(M\), is distance non-decreasing. A space metric \(M\) is said to have nonpositive curvature if it locally satisfies the CAT(0) condition. In this paper, the authors prove that if \(M\) is a Riemannian manifold of constant nonpositive sectional curvature, then it admits a piecewise Euclidean structure of nonpositive curvature. To prove the result, they make use of the quadratic form model of hyperbolic space, Voronoi diagrams, and Delaunay tesselations. The same problem, when the curvature of \(M\) is nonpositive but not necessarily constant, remains open.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 1046977
zbMATH Keywords: Alexandrov space
zbMATH Keywords: negative curvature
zbMATH Keywords: piecewise Euclidean
zbMATH Keywords: Dirichlet region
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: Voronoi diagram
author: Ruth Charney
author: Michael W. Davis
author: Gábor Moussong
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005630
OpenAlex ID: W2051726905
DOI: 10.1307/MMJ/1029005630
description: scientific article; zbMATH DE number 1046977