Fenchel–Nielsen coordinates
E898483
Fenchel–Nielsen coordinates are a system of parameters that describe hyperbolic structures on surfaces by recording lengths and twist angles along a collection of simple closed geodesics.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Teichmüller theory concept
ⓘ
coordinate system ⓘ hyperbolic geometry concept ⓘ parameterization ⓘ |
| appliesTo |
Riemann surfaces of genus g ≥ 2
ⓘ
hyperbolic surfaces ⓘ oriented surfaces of finite type ⓘ |
| assumes |
complete hyperbolic metric of finite area
ⓘ
geodesic boundary or cusps ⓘ |
| basedOn | pants decomposition of a surface ⓘ |
| codomain | Euclidean space ⓘ |
| defines |
length parameters
ⓘ
twist parameters ⓘ |
| dimensionMatches | dimension of Teichmüller space ⓘ |
| domain | Teichmüller space of a surface NERFINISHED ⓘ |
| forSurfaceOfGenus | g with n boundary components ⓘ |
| generalizes | Fenchel’s description of hyperbolic polygons ⓘ |
| hasAlternativeFormulation | complex Fenchel–Nielsen coordinates ⓘ |
| hasComponent |
length coordinate
ⓘ
twist coordinate ⓘ |
| lengthParameterRepresents | hyperbolic length of a simple closed geodesic ⓘ |
| namedAfter |
Jakob Nielsen
NERFINISHED
ⓘ
Werner Fenchel NERFINISHED ⓘ |
| numberOfLengthParameters | 3g - 3 + n GENERATED ⓘ |
| numberOfTwistParameters | 3g - 3 + n GENERATED ⓘ |
| property |
coordinates are real-valued
ⓘ
depend on choice of pants decomposition ⓘ give a global real-analytic homeomorphism from Teichmüller space to Euclidean space ⓘ |
| relatedTo |
Weil–Petersson symplectic form
NERFINISHED
ⓘ
mapping class group NERFINISHED ⓘ moduli space of Riemann surfaces ⓘ |
| totalNumberOfParameters | 6g - 6 + 2n ⓘ |
| twistParameterMeasuredAlong | simple closed geodesic ⓘ |
| twistParameterRepresents | relative twist of pairs of pants along a geodesic ⓘ |
| usedFor |
describing hyperbolic structures on surfaces
ⓘ
describing metrics of constant negative curvature ⓘ parameterizing Teichmüller space ⓘ studying moduli of Riemann surfaces ⓘ |
| usedIn |
complex analysis on Riemann surfaces
ⓘ
geometric group theory ⓘ low-dimensional topology ⓘ study of Kleinian groups ⓘ |
| usesConcept |
Dehn twist
NERFINISHED
ⓘ
geodesic length ⓘ hyperbolic metric ⓘ simple closed geodesics ⓘ twist parameter ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.