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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Poincaré half-plane model | 2 |
| Poincaré disk model | 1 |
| Poincaré half-plane | 1 |
| Poincaré upper half-plane model canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338662 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Poincaré upper half-plane model Context triple: [Farey tessellation, embeddedIn, Poincaré upper half-plane model]
-
A.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
B.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
E.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Poincaré upper half-plane model Target entity description: 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.
-
A.
Riemann sphere
The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
-
B.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
E.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
- F. None of above. chosen
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
ⓘ
linked to:
Poincaré upper half-plane model
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)
ⓘ
linked to:
PSL(2,ℝ)
|
| 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) ⓘ |
| hasMetric | ds² = (dx² + dy²) / y² ⓘ |
| hasMetricTensor | g = (1/y²)(dx⊗dx + dy⊗dy) ⓘ |
| hasNaturalActionBy | Möbius transformations with real coefficients ⓘ |
| hasOrientationPreservingIsometryGroup | PSL(2,ℝ) ⓘ |
| 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 ⓘ |
| isConformalTo | Euclidean upper half-plane ⓘ |
| isEquivalentTo | upper half-plane with hyperbolic metric ⓘ |
| isHomogeneous | true ⓘ |
| isIsometricTo | Poincaré disk model ⓘ |
| isIsotropic | true ⓘ |
| isModelOf |
Lobachevskian geometry
ⓘ
linked to:
Non-Euclidean Geometry
|
| isSimplyConnected | true ⓘ |
| isSimplyTransitiveUnder | PSL(2,ℝ) on oriented geodesics ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| represents | hyperbolic plane ⓘ |
| usedIn |
Fuchsian groups
ⓘ
linked to:
Fuchsian group
Kleinian groups ⓘ
linked to:
Kleinian group
Teichmüller theory ⓘ complex analysis ⓘ hyperbolic geometry ⓘ modular forms theory ⓘ number theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Poincaré upper half-plane model Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Poincaré upper half-plane model
subject linked to:
Non-Euclidean Geometry
linked to: Poincaré upper half-plane model
linked to: Poincaré upper half-plane model
linked to: Poincaré upper half-plane model