Metrics on hyperbolic convex polyhedra
type: Item
instance of: scholarly article
title: Metrics on hyperbolic convex polyhedra
zbMATH Open document ID: 0912.52008
author: Jean-Marc Schlenker
published in: Journal of Differential Geometry
publication date: 1999-05-16T00:00:00Z
review text: There is a traditional way of defining a metric for the projective model of hyperbolic space: For a unit ball \(C \subset R^n\) given two interior points \(a,b\) one considers the usual line through \(a,b\) and finds the intersections \(x,y\) of this line with the boundary of \(C\). The Hilbert distance between \(a\) and \(b\) is simply \(-\frac{1}{2} \log(\frac{xa}{xb} \frac{by}{ay})\). Thinking of \(\partial C\) as a quadric not as a convex set one can generalize the above formula for points in \(R^n -C\). The ``distance'' becomes then a certain complex number. One can for example speak of the distance between two points outside \(C\). The new space with this ``distance'' is called the Sitter hyperbolic space, denoted \(HS^n\) and it has natural notions of geodesic and polyhedron. Sitter spaces have also a notion of duality between points and hyperplanes. The main goal of this article is to show that several well-known results about polyhedra in the classical hyperbolic space extend to Sitter spaces. The author presents generalizations of results by Alexandrov, Andreev, Pogorelov and Rivin.
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 52B12
zbMATH DE Number: 1203424
zbMATH Keywords: Sitter spaces
zbMATH Keywords: convexity in hyperbolic space
zbMATH Keywords: hyperbolic polyhedra
zbMATH Keywords: Hilbert metric
zbMATH Keywords: compact convex polyhedra
zbMATH Keywords: hyperbolic 3-space
Wikidata QID: Q115174729
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214460799
OpenAlex ID: W1547598092
DOI: 10.4310/JDG/1214460799
description: scientific article; zbMATH DE number 1203424