A characterization of compact convex polyhedra in hyperbolic 3-space
type: Item
instance of: scholarly article
title: A characterization of compact convex polyhedra in hyperbolic 3-space
zbMATH Open document ID: 0784.52013
DOI: 10.1007/BF01231281
author: Craig Hodgson
author: Igor Rivin
published in: Inventiones Mathematicae
publication date: 1994-03-27T00:00:00Z
full work available at URL: https://eudml.org/doc/144071
review text: This paper is a comprehensive survey with detailed proofs based on the Ph. D. dissertation of the second author. \textit{A. D. Alexandrov}, Convex Polyhedra (in Russian), Moscow, GITTL, 1950, (German translation: Berlin, Akademie Verlag 1958) has given a complete characterization of the shapes of compact convex polyhedra in the hyperbolic 3-space \(H^ 3\) in terms of the intrinsic hyperbolic metric on the boundary. The authors give an analogous extrinsic description in terms of their dihedral angles. The Gauss map of a compact convex polyhedron on the Euclidean 3-space \(E^ 3\) has been extended onto \(H^ 3\) by using the hyperboloid model of \(H^ 3\). The hyperbolic polar image \(G(P)\) of \(P\) has a realization as a ``convex polyhedron'' in the ``de Sitter sphere'' \(S^ 2_ 1\). The main result is Theorem 1.1. A metric space \((M,g)\) homeomorphic to \(S^ 2\) can arise as the Gaussian image \(G(P)\) of a compact convex polyhedron \(P\) in \(H^ 3\) iff the following conditions hold: (a) The metric \(g\) has constant curvature 1 away from a finite collection of cone points \(c_ i\). (b) The cone angle at each \(c_ i\) is greater than \(2\pi\). (c) The lengths of closed geodesics of \((M,g)\) are strictly greater than \(2\pi\). These results have interesting consequences, e.g. Andreev's characterization of compact convex polyhedra in \(H^ 3\) with dihedral angles not greater than \(\pi/2\) follows very easily [see the first author, Ohio State Univ. Math. Res. Inst. Publ. 1, 185-193 (1992; Zbl 0765.52013)]. Another result states that the face angles of a convex polyhedron \(P\) in \(H^ 3\) determine it up to congruence. The second author applied the current approach to characterize convex polyhedra of finite volume just in the following review: [\textit{I. Rivin}: On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32, No. 1, 87-92 (1993; see the paper below)]. In the projective Cayley-Klein model of \(H^ 3\), an ideal convex hyperbolic polyhedron is…
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 179516
zbMATH Keywords: survey
zbMATH Keywords: compact convex polyhedra
zbMATH Keywords: hyperbolic 3-space
Wikidata QID: Q56565595
reviewed by: Emil Molnár
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 179516