Fenchel–Nielsen coordinates
Encyclopedia
In mathematics, Fenchel–Nielsen coordinates are coordinates for Teichmüller space
introduced by Werner Fenchel
and Jakob Nielsen
.
g > 1. The Fenchel–Nielsen coordinates depend on a choice of 6g − 6 curves on S, as follows. The Riemann surface S can be divided up into 2g − 2 pairs of pants by cutting along 3g − 3 disjoint simple closed curves. For each of these 3g − 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 3g − 3 positive real numbers called the lengths and 3g − 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 3g − 3 disjoint simple closed curves.
The twists of the Fenchel–Nielsen coordinates are given as follows. There is one twist for each of the 3g − 3 curves crossing one of the 3g − 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.
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
introduced by Werner Fenchel
Werner Fenchel
Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also...
and Jakob Nielsen
Jakob Nielsen (mathematician)
Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium...
.
Definition
Suppose that S is a compact Riemann surface of genusGenus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g > 1. The Fenchel–Nielsen coordinates depend on a choice of 6g − 6 curves on S, as follows. The Riemann surface S can be divided up into 2g − 2 pairs of pants by cutting along 3g − 3 disjoint simple closed curves. For each of these 3g − 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 3g − 3 positive real numbers called the lengths and 3g − 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 3g − 3 disjoint simple closed curves.
The twists of the Fenchel–Nielsen coordinates are given as follows. There is one twist for each of the 3g − 3 curves crossing one of the 3g − 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.