|
In mathematics, Fenchel–Nielsen coordinates are coordinates for Teichmüller space introduced by Werner Fenchel and Jakob Nielsen. ==Definition== Suppose that ''S'' is a compact Riemann surface of genus ''g'' > 1. The Fenchel–Nielsen coordinates depend on a choice of 6''g'' − 6 curves on ''S'', as follows. The Riemann surface ''S'' can be divided up into 2''g'' − 2 pairs of pants by cutting along 3''g'' − 3 disjoint simple closed curves. For each of these 3''g'' − 3 curves γ, choose an arc crossing it that ends in other boundary components of the pairs of pants with boundary containing γ. The Fenchel–Nielsen coordinates for a point of the Teichmüller space of ''S'' consist of 3''g'' − 3 positive real numbers called the lengths and 3''g'' − 3 real numbers called the twists. A point of Teichmüller space is represented by a hyperbolic metric on ''S''. The lengths of the Fenchel–Nielsen coordinates are the lengths of geodesics homotopic to the 3''g'' − 3 disjoint simple closed curves. The twists of the Fenchel–Nielsen coordinates are given as follows. There is one twist for each of the 3''g'' − 3 curves crossing one of the 3''g'' − 3 disjoint simple closed curves γ. Each of these is homotopic to a curve that consists of 3 geodesic segments, the middle one of which follows the geodesic of γ. The twist is the (positive or negative) distance the middle segment travels along the geodesic of γ. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fenchel–Nielsen coordinates」の詳細全文を読む スポンサード リンク
|