Uniformization theorem (original) (raw)

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En mathématiques, le théorème d'uniformisation de Riemann est un résultat de base dans la théorie des surfaces de Riemann, c'est-à-dire des variétés complexes de dimension 1. Il assure que toute surface de Riemann simplement connexe peut être mise en correspondance biholomorphe avec l'une des trois surfaces suivantes : le plan complexe C, le disque unité de ce plan, ou la sphère de Riemann, c'est-à-dire la droite projective complexe P1(C).

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dbo:abstract En mathématiques, le théorème d'uniformisation de Riemann est un résultat de base dans la théorie des surfaces de Riemann, c'est-à-dire des variétés complexes de dimension 1. Il assure que toute surface de Riemann simplement connexe peut être mise en correspondance biholomorphe avec l'une des trois surfaces suivantes : le plan complexe C, le disque unité de ce plan, ou la sphère de Riemann, c'est-à-dire la droite projective complexe P1(C). (fr) Il teorema di uniformizzazione di Riemann è un importante teorema di analisi complessa, dimostrato dal matematico Bernhard Riemann. Il teorema descrive un forte collegamento fra l'analisi complessa e la geometria differenziale per le superfici. (it) 一意化定理(uniformization theorem)とは、すべての単連結リーマン面は、開円板、複素平面、リーマン球面の 3つのうちのひとつに共形同値であるという定理である。特に、単連結リーマン面は(constant curvature)のリーマン計量を持つ。この定理は普遍被覆リーマン面を楕円型（正の曲率、正の曲がった曲率をもつ）、放物型（平坦）、双曲型(負曲率）として分類する。 一意化定理はリーマンの写像定理の平面の固有な単連結開部分集合から、任意の単連結はリーマン面への一般化である。 一意化定理は、任意の連結である第二可算の面の同様な結果、定数曲率のリーマン計量を与えることができることを意味している。 (ja) In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. (en) 복소해석학에서 균일화 정리(均一化定理, uniformization theorem)는 단일 연결 리만 곡면이 열린 단위 원판이나 복소평면, 리만 구 가운데 하나로 전단사 등각 사상이 존재한다는 정리다. (ko) Теорема об униформизации — обобщение теоремы Римана об отображении на двумерные римановы многообразия. Можно сказать, что теорема даёт наилучшую метрику в данном конформном классе. (ru) In de riemann-meetkunde, een deelgebied van de wiskunde, zegt de uniformeringsstelling dat elk enkelvoudig samenhangende riemann-oppervlak hoekgetrouw equivalent is aan een van de drie domeinen: de open eenheidsschijf, het complexe vlak of de riemann-sfeer. In het bijzonder staat het een riemann-metriek met constante kromming toe. Dit classificeert riemann-oppervlakken als elliptisch (positief gekromd - of beter een constante positieve metriek toelatend), parabolisch (vlak) of hyperbolisch (negatief gekromd) op basis van hun universele overdekking. De uniformeringsstelling is een veralgemening van de afbeeldingstelling van Riemann voor enkelvoudig samenhangende open deelverzamelingen van het vlak naar willekeurige enkelvoudig samenhangende riemann-oppervlakken. De uniformeringsstelling impliceert een soortgelijk resultaat voor willekeurig samenhangende tweedst-aftelbare oppervlakken: men kan zij uitrusten met een riemann-metriek met constante kromming. (nl) Теорема про уніформізацію — узагальнення теореми Рімана про відображення на двовимірні ріманові многовиди. Можна сказати, що теорема дає найкращу метрику в даному конформному класі. (uk) 数学上，曲面的单值化定理是说任何曲面上都有一个常高斯曲率的度量。事实上，在每一个给定的中我们都可以找到一个常高斯曲率的度量。等价的說，用复分析的语言，任何单连通的黎曼曲面都共形等价於复平面、单位圆盘和黎曼球面三者之一。 (zh)
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dbp:authorlink Paul Koebe (en)
dbp:first Paul (en) N.A. (en)
dbp:id U/u095290 (en)
dbp:last Gusevskii (en) Koebe (en)
dbp:title Uniformization (en)
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rdfs:comment En mathématiques, le théorème d'uniformisation de Riemann est un résultat de base dans la théorie des surfaces de Riemann, c'est-à-dire des variétés complexes de dimension 1. Il assure que toute surface de Riemann simplement connexe peut être mise en correspondance biholomorphe avec l'une des trois surfaces suivantes : le plan complexe C, le disque unité de ce plan, ou la sphère de Riemann, c'est-à-dire la droite projective complexe P1(C). (fr) Il teorema di uniformizzazione di Riemann è un importante teorema di analisi complessa, dimostrato dal matematico Bernhard Riemann. Il teorema descrive un forte collegamento fra l'analisi complessa e la geometria differenziale per le superfici. (it) 一意化定理(uniformization theorem)とは、すべての単連結リーマン面は、開円板、複素平面、リーマン球面の 3つのうちのひとつに共形同値であるという定理である。特に、単連結リーマン面は(constant curvature)のリーマン計量を持つ。この定理は普遍被覆リーマン面を楕円型（正の曲率、正の曲がった曲率をもつ）、放物型（平坦）、双曲型(負曲率）として分類する。 一意化定理はリーマンの写像定理の平面の固有な単連結開部分集合から、任意の単連結はリーマン面への一般化である。 一意化定理は、任意の連結である第二可算の面の同様な結果、定数曲率のリーマン計量を与えることができることを意味している。 (ja) 복소해석학에서 균일화 정리(均一化定理, uniformization theorem)는 단일 연결 리만 곡면이 열린 단위 원판이나 복소평면, 리만 구 가운데 하나로 전단사 등각 사상이 존재한다는 정리다. (ko) Теорема об униформизации — обобщение теоремы Римана об отображении на двумерные римановы многообразия. Можно сказать, что теорема даёт наилучшую метрику в данном конформном классе. (ru) Теорема про уніформізацію — узагальнення теореми Рімана про відображення на двовимірні ріманові многовиди. Можна сказати, що теорема дає найкращу метрику в даному конформному класі. (uk) 数学上，曲面的单值化定理是说任何曲面上都有一个常高斯曲率的度量。事实上，在每一个给定的中我们都可以找到一个常高斯曲率的度量。等价的說，用复分析的语言，任何单连通的黎曼曲面都共形等价於复平面、单位圆盘和黎曼球面三者之一。 (zh) In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. (en) In de riemann-meetkunde, een deelgebied van de wiskunde, zegt de uniformeringsstelling dat elk enkelvoudig samenhangende riemann-oppervlak hoekgetrouw equivalent is aan een van de drie domeinen: de open eenheidsschijf, het complexe vlak of de riemann-sfeer. In het bijzonder staat het een riemann-metriek met constante kromming toe. Dit classificeert riemann-oppervlakken als elliptisch (positief gekromd - of beter een constante positieve metriek toelatend), parabolisch (vlak) of hyperbolisch (negatief gekromd) op basis van hun universele overdekking. (nl)
rdfs:label Théorème d'uniformisation de Riemann (fr) Teorema di uniformizzazione di Riemann (it) 균일화 정리 (ko) 一意化定理 (ja) Uniformeringsstelling (nl) Uniformization theorem (en) Теорема об униформизации (ru) 单值化定理 (zh) Теорема уніформізації (uk)
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