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In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following: * the Riemann sphere, * the complex plane, * the unit disk D equivalently the upper half-plane H. In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere.

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  • Fundamentalpolygon
  • Fundamental polygon
  • 基本多边形
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  • In der Mathematik kann jede im topologischen Sinn geschlossene Fläche erzeugt werden, indem man die Seiten eines Polygons mit gerader Seitenanzahl paarweise identifiziert. Dieses Polygon nennt man Fundamentalpolygon. Diese Polygone kann man durch eine Zeichenkette beschreiben, die jeder Seite ein Symbol zuordnet. Seiten, die miteinander identifiziert werden erhalten dabei das gleiche Symbol. Ein zusätzlicher Exponent 1 oder −1 gibt die Orientierung der Seite an.
  • 在数学上,每个闭曲面在几何拓扑的意义下,可以由一个偶数条边的有向多边形,把它的边成对地粘合构造出来,这样的多边形称之为基本多边形(fundamental polygon)。 这个构造可以表示成一个长为2n的字符串,一共n个不同的符号,每个符号出现两次带有指数 +1或 -1。指数 -1的符号对应于该边的定向与基本多边形的定向相反。
  • In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following: * the Riemann sphere, * the complex plane, * the unit disk D equivalently the upper half-plane H. In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere.
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