Poincaré metric
E898491
Riemannian metric
complete metric
conformal metric
hyperbolic metric
metric of constant negative curvature
The Poincaré metric is the canonical complete Riemannian metric of constant negative curvature on simply connected Riemann surfaces like the unit disk or upper half-plane, fundamental in complex analysis and hyperbolic geometry.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Riemannian metric
ⓘ
complete metric ⓘ conformal metric ⓘ hyperbolic metric ⓘ metric of constant negative curvature ⓘ |
| belongsTo |
Poincaré disk model
NERFINISHED
ⓘ
Poincaré half-plane model NERFINISHED ⓘ |
| characterizes | hyperbolic Riemann surfaces ⓘ |
| definedOn |
simply connected Riemann surfaces
ⓘ
unit disk ⓘ upper half-plane ⓘ |
| extendsTo | universal cover of any hyperbolic Riemann surface ⓘ |
| geodesicsOnUnitDisk | circles and lines orthogonal to unit circle ⓘ |
| geodesicsOnUpperHalfPlane | vertical lines and semicircles orthogonal to real axis ⓘ |
| hasBoundaryAtInfinity |
extended real line
ⓘ
unit circle ⓘ |
| hasCurvature | -1 ⓘ |
| hasDimension | 2 ⓘ |
| hasLineElementOnUnitDisk | 4|dz|^2/(1-|z|^2)^2 ⓘ |
| hasLineElementOnUpperHalfPlane | |dz|^2/(Im z)^2 ⓘ |
| hasSectionalCurvature |
-1 everywhere
ⓘ
constant negative ⓘ |
| induces |
geodesics as circular arcs orthogonal to boundary
ⓘ
hyperbolic distance ⓘ |
| isCanonical | true ⓘ |
| isComplete | true ⓘ |
| isCompleteOn |
unit disk
ⓘ
upper half-plane ⓘ |
| isConformallyEquivalentVia | Cayley transform between disk and upper half-plane NERFINISHED ⓘ |
| isConformalTo | Euclidean metric ⓘ |
| isEquivalentTo |
Carathéodory metric on the unit disk
NERFINISHED
ⓘ
Kobayashi metric on the unit disk NERFINISHED ⓘ |
| isInvariantUnder |
Möbius transformations preserving the domain
ⓘ
PSL(2,R) NERFINISHED ⓘ SU(1,1) NERFINISHED ⓘ biholomorphic automorphisms ⓘ |
| isMaximalAmong | conformal metrics of curvature ≤ -1 on the disk ⓘ |
| isModelOf | two-dimensional hyperbolic geometry ⓘ |
| isUniqueUpTo | biholomorphic equivalence ⓘ |
| isUsedToDefine |
hyperbolic distance on the disk
ⓘ
hyperbolic distance on the upper half-plane ⓘ |
| namedAfter | Henri Poincaré NERFINISHED ⓘ |
| usedIn |
Kobayashi hyperbolicity
NERFINISHED
ⓘ
Teichmüller theory NERFINISHED ⓘ complex analysis ⓘ differential geometry ⓘ geometric function theory ⓘ geometric group theory ⓘ hyperbolic geometry ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.