Hyperbolic manifold (original) (raw)
In der Mathematik sind hyperbolische Mannigfaltigkeiten Riemannsche Mannigfaltigkeiten mit konstanter negativer Schnittkrümmung. Sie spielen eine wichtige Rolle in der niedrig-dimensionalen Topologie, insbesondere in Thurstons Geometrisierungsprogramm.
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| dbo:abstract | In der Mathematik sind hyperbolische Mannigfaltigkeiten Riemannsche Mannigfaltigkeiten mit konstanter negativer Schnittkrümmung. Sie spielen eine wichtige Rolle in der niedrig-dimensionalen Topologie, insbesondere in Thurstons Geometrisierungsprogramm. (de) In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. (en) En mathématiques, une variété hyperbolique est un espace dans lequel chaque point apparaît localement comme espace hyperbolique d'une certaine dimension. Ces variétés sont spécifiquement étudiées en dimensions 2 et 3, où elles sont appelées respectivement surfaces de Riemann et (en). Dans ces dimensions, elles sont importantes parce que la plupart des variétés peuvent être transformées en variétés hyperboliques par homéomorphisme. C'est une conséquence du théorème d'uniformisation de Riemann pour les surfaces et de la conjecture de géométrisation de Thurston, prouvée par Grigori Perelman, pour les 3-variétés. (fr) 数学において双曲多様体（そうきょくたようたい、英: hyperbolic manifold）とは、すべての点が局所的にはある次元のであるように見える空間（＝可微分多様体）のことを言う。特に 2 次元および 3 次元において研究され、そのような場合には双曲曲面および双曲3次元多様体とそれぞれ呼ばれる。それらの次元においてこの多様体が重要となる理由として、殆どの多様体は位相同型によって双曲多様体に作り変えることが出来る、という点が挙げられる。これは曲面に対する一意化定理や、ペレルマンによって証明された 3 次元多様体に対する幾何化定理の帰結である。 (ja) In geometria, una varietà iperbolica è una varietà riemanniana avente curvatura sezionale ovunque -1. Se la varietà è completa, questa ha come rivestimento universale lo spazio iperbolico . Esempi di varietà iperboliche sono le superfici aventi caratteristica di Eulero negativa (dotate di un tensore metrico opportuno). Anche molte 3-varietà sono varietà iperboliche. (it) |
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| rdfs:comment | In der Mathematik sind hyperbolische Mannigfaltigkeiten Riemannsche Mannigfaltigkeiten mit konstanter negativer Schnittkrümmung. Sie spielen eine wichtige Rolle in der niedrig-dimensionalen Topologie, insbesondere in Thurstons Geometrisierungsprogramm. (de) In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. (en) En mathématiques, une variété hyperbolique est un espace dans lequel chaque point apparaît localement comme espace hyperbolique d'une certaine dimension. Ces variétés sont spécifiquement étudiées en dimensions 2 et 3, où elles sont appelées respectivement surfaces de Riemann et (en). Dans ces dimensions, elles sont importantes parce que la plupart des variétés peuvent être transformées en variétés hyperboliques par homéomorphisme. C'est une conséquence du théorème d'uniformisation de Riemann pour les surfaces et de la conjecture de géométrisation de Thurston, prouvée par Grigori Perelman, pour les 3-variétés. (fr) 数学において双曲多様体（そうきょくたようたい、英: hyperbolic manifold）とは、すべての点が局所的にはある次元のであるように見える空間（＝可微分多様体）のことを言う。特に 2 次元および 3 次元において研究され、そのような場合には双曲曲面および双曲3次元多様体とそれぞれ呼ばれる。それらの次元においてこの多様体が重要となる理由として、殆どの多様体は位相同型によって双曲多様体に作り変えることが出来る、という点が挙げられる。これは曲面に対する一意化定理や、ペレルマンによって証明された 3 次元多様体に対する幾何化定理の帰結である。 (ja) In geometria, una varietà iperbolica è una varietà riemanniana avente curvatura sezionale ovunque -1. Se la varietà è completa, questa ha come rivestimento universale lo spazio iperbolico . Esempi di varietà iperboliche sono le superfici aventi caratteristica di Eulero negativa (dotate di un tensore metrico opportuno). Anche molte 3-varietà sono varietà iperboliche. (it) |
| rdfs:label | Hyperbolische Mannigfaltigkeit (de) Varietà iperbolica (it) Variété hyperbolique (fr) Hyperbolic manifold (en) 双曲多様体 (ja) |
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