Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds
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.
scientific article; zbMATH DE number 1046977