Poincaré upper half-plane model
E656696
2-dimensional manifold
Riemannian manifold
conformal model of the hyperbolic plane
model of hyperbolic geometry
simply connected surface
The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Poincaré disk model | 1 |
| Poincaré half-plane model | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
2-dimensional manifold
ⓘ
Riemannian manifold ⓘ conformal model of the hyperbolic plane ⓘ model of hyperbolic geometry ⓘ simply connected surface ⓘ |
| alsoKnownAs |
Poincaré half-plane
NERFINISHED
ⓘ
upper half-plane model ⓘ |
| boundaryAtInfinity | extended real line ℝ ∪ {∞} ⓘ |
| hasConditionOnImaginaryPart | y > 0 ⓘ |
| hasCurvatureNormalization | constant curvature -1 ⓘ |
| hasDimension | 2 ⓘ |
| hasDistanceElement | ds = √(dx² + dy²)/y ⓘ |
| hasFullIsometryGroup | PGL(2,ℝ) NERFINISHED ⓘ |
| hasGaussianCurvature | -1 ⓘ |
| hasGeodesicBoundaryCondition | geodesics meet real axis orthogonally ⓘ |
| hasGeodesics |
semicircles orthogonal to the real axis
ⓘ
vertical lines ⓘ |
| hasGeodesicSymmetry | reflections in geodesics are isometries ⓘ |
| hasIsometryGroup | PSL(2,ℝ) NERFINISHED ⓘ |
| hasMetric | ds² = (dx² + dy²) / y² ⓘ |
| hasMetricTensor | g = (1/y²)(dx⊗dx + dy⊗dy) ⓘ |
| hasNaturalActionBy | Möbius transformations with real coefficients ⓘ |
| hasOrientationPreservingIsometryGroup | PSL(2,ℝ) NERFINISHED ⓘ |
| hasOrientationReversingIsometries | complex conjugation composed with PSL(2,ℝ) ⓘ |
| hasSectionalCurvature | -1 ⓘ |
| hasStandardCoordinate | z = x + i y ⓘ |
| hasTopology | standard subspace topology from ℂ ⓘ |
| hasUnderlyingSet | {z ∈ ℂ : Im(z) > 0} ⓘ |
| hasVolumeElement | dA = dx dy / y² ⓘ |
| isComplete | true ⓘ |
| isConformallyEquivalentTo | Poincaré disk model NERFINISHED ⓘ |
| isConformalTo | Euclidean upper half-plane NERFINISHED ⓘ |
| isEquivalentTo | upper half-plane with hyperbolic metric ⓘ |
| isHomogeneous | true ⓘ |
| isIsometricTo | Poincaré disk model NERFINISHED ⓘ |
| isIsotropic | true ⓘ |
| isModelOf | Lobachevskian geometry NERFINISHED ⓘ |
| isSimplyConnected | true ⓘ |
| isSimplyTransitiveUnder | PSL(2,ℝ) on oriented geodesics ⓘ |
| namedAfter | Henri Poincaré NERFINISHED ⓘ |
| represents | hyperbolic plane ⓘ |
| usedIn |
Fuchsian groups
NERFINISHED
ⓘ
Kleinian groups NERFINISHED ⓘ Teichmüller theory NERFINISHED ⓘ complex analysis ⓘ hyperbolic geometry ⓘ modular forms theory ⓘ number theory ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Non-Euclidean geometry
this entity surface form:
Poincaré half-plane model
this entity surface form:
Poincaré disk model