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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Poincaré metric canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991775 — 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 metric Context triple: [uniformization theorem, relatesTo, Poincaré metric]
-
A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
B.
Poincaré upper half-plane model
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.
-
C.
Bergman metric
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
-
D.
Kobayashi metric
The Kobayashi metric is an intrinsic pseudometric in complex analysis that measures hyperbolic distance on complex manifolds and generalizes the Poincaré metric to higher dimensions.
-
E.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
- 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 metric Target entity description: 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.
-
A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
B.
Poincaré upper half-plane model
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.
-
C.
Bergman metric
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
-
D.
Kobayashi metric
The Kobayashi metric is an intrinsic pseudometric in complex analysis that measures hyperbolic distance on complex manifolds and generalizes the Poincaré metric to higher dimensions.
-
E.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Riemannian metric
ⓘ
complete metric ⓘ conformal metric ⓘ hyperbolic metric ⓘ metric of constant negative curvature ⓘ |
| belongsTo |
Poincaré disk model
ⓘ
Poincaré half-plane model ⓘ
linked to:
Poincaré upper half-plane model
|
| 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
ⓘ
linked to:
Cayley transform
|
| isConformalTo | Euclidean metric ⓘ |
| isEquivalentTo |
Carathéodory metric on the unit disk
ⓘ
linked to:
Carathéodory metric
Kobayashi metric on the unit disk ⓘ
linked to:
Kobayashi metric
|
| isInvariantUnder |
Möbius transformations preserving the domain
ⓘ
PSL(2,R) ⓘ
linked to:
PSL(2,ℝ)
SU(1,1) ⓘ 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 ⓘ |
| usedIn |
Kobayashi hyperbolicity
ⓘ
Teichmüller theory ⓘ complex analysis ⓘ differential geometry ⓘ geometric function theory ⓘ geometric group theory ⓘ hyperbolic geometry ⓘ |
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 metric Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.