Topic
Hyperbolic space
About: Hyperbolic space is a research topic. Over the lifetime, 4045 publications have been published within this topic receiving 67810 citations. The topic is also known as: Lobachevsky space & hyperbolic spaces.
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Book•
09 May 1983
TL;DR: In this article, the authors describe the geometric theory of discrete groups and associated tesselations of the underlying space, and develop the theory of Mobius transformations in n-dimensional Euclidean space.
Abstract: Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean space. These transformations are discussed as isometries of hyperbolic space and are then identified with the elementary transformations of complex analysis. A detailed account of analytic hyperbolic trigonometry is given, and this forms the basis of the subsequent analysis of tesselations of the hyperbolic plane. Emphasis is placed on the geometrical aspects of the subject and on the universal constraints which must be satisfied by all tesselations.
1,803 citations
TL;DR: In the case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in a surface is isometric to a neighborhood in the hyper-bolic plane, and the surface itself is the quotient of the hyperbola by a discrete group of motions as discussed by the authors.
Abstract: 1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the
1,641 citations
Book•
01 Jan 1984
1,325 citations
01 Jan 1992
TL;DR: In this article, the authors provide an exposition of some fundamental results of hyperbolic manifolds, while being as self-contained, complete, detailed and unified as possible, and much space is devoted to the 3D case, based on the representation of three manifolds as glued ideal tetrahedra.
Abstract: Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible. Following some classical material on the hyperbolic space and the Teichmuller space, the book centers on the two fundamental results: Mostow's rigidity theorem (including a complete proof, following Gromov and Thurston) and Margulis' lemma. These then form the basis for studying Chabauty and geometric topology; a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory; and much space is devoted to the 3D case: a complete and elementary proof of the hyperbolic surgery theorem, based on the representation of three manifolds as glued ideal tetrahedra.
874 citations
TL;DR: In this paper, the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Abstract: © Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
741 citations