Kobayashi metric
E521038
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kobayashi metric canonical | 6 |
| Kobayashi metric on the unit disk | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5446358 — 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: Kobayashi metric Context triple: [Carathéodory metric, majorizedBy, Kobayashi metric]
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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.
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B.
Gödel metric
The Gödel metric is a solution to Einstein's field equations that describes a rotating universe allowing for closed timelike curves and thus the theoretical possibility of time travel.
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C.
Kerr metric
The Kerr metric is the exact general relativity solution describing the spacetime geometry around a rotating, uncharged black hole.
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D.
Reissner–Nordström metric
The Reissner–Nordström metric is an exact solution in general relativity describing the spacetime geometry outside a static, spherically symmetric, electrically charged black hole.
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E.
Kerr–Schild coordinates
Kerr–Schild coordinates are a coordinate system used to express the Kerr spacetime metric in a form that highlights its structure as a perturbation of flat Minkowski space along a principal null direction.
- 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: Kobayashi metric Target entity description: 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.
-
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.
Gödel metric
The Gödel metric is a solution to Einstein's field equations that describes a rotating universe allowing for closed timelike curves and thus the theoretical possibility of time travel.
-
C.
Kerr metric
The Kerr metric is the exact general relativity solution describing the spacetime geometry around a rotating, uncharged black hole.
-
D.
Reissner–Nordström metric
The Reissner–Nordström metric is an exact solution in general relativity describing the spacetime geometry outside a static, spherically symmetric, electrically charged black hole.
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E.
Kerr–Schild coordinates
Kerr–Schild coordinates are a coordinate system used to express the Kerr spacetime metric in a form that highlights its structure as a perturbation of flat Minkowski space along a principal null direction.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
hyperbolic metric
ⓘ
intrinsic metric ⓘ invariant metric in complex analysis ⓘ pseudometric ⓘ |
| agreesWith | Hilbert metric on the unit ball in C^n up to equivalence ⓘ |
| appearsIn | "Hyperbolic Manifolds and Holomorphic Mappings" by Shoshichi Kobayashi ⓘ |
| appliesTo |
complex analytic spaces
ⓘ
complex manifolds ⓘ |
| coincidesWith |
Carathéodory metric on the unit disc
ⓘ
Poincaré metric on the unit disc ⓘ |
| definedUsing |
chains of holomorphic discs
ⓘ
infimum of lengths of chains of analytic discs ⓘ |
| field |
complex analysis
ⓘ
complex differential geometry ⓘ several complex variables ⓘ |
| generalizes | Poincaré metric ⓘ |
| introducedBy | Shoshichi Kobayashi ⓘ |
| introducedIn | 1960s ⓘ |
| isCompleteOn |
bounded convex domains in C^n
ⓘ
bounded strongly pseudoconvex domains ⓘ |
| isContractedBy | holomorphic self-maps ⓘ |
| isDistanceDecreasingUnderHolomorphicMaps | true ⓘ |
| isEquivalentTo |
Bergman metric on certain bounded symmetric domains
ⓘ
Carathéodory metric on taut domains ⓘ |
| isFinslerType | true ⓘ |
| isHolomorphicallyInvariant | true ⓘ |
| isIntrinsic | true ⓘ |
| isInvariantUnder | biholomorphic maps ⓘ |
| isLargestPseudometricWithDistanceDecreasingProperty | true ⓘ |
| isLocalFinslerMetricOn | tangent bundle of a complex manifold ⓘ |
| isMonotoneUnder | holomorphic mappings ⓘ |
| isNondegenerateOn | Kobayashi hyperbolic manifolds ⓘ |
| isPseudometric | true ⓘ |
| isToolFor |
proving Brody hyperbolicity
ⓘ
studying negative curvature in complex geometry ⓘ |
| isUpperSemicontinuous | true ⓘ |
| mayFailToSeparatePoints | true ⓘ |
| namedAfter | Shoshichi Kobayashi ⓘ |
| relatedConcept |
Bergman metric
ⓘ
Carathéodory metric ⓘ Kobayashi hyperbolicity ⓘ Teichmüller metric ⓘ |
| satisfies | Schwarz–Pick type inequalities ⓘ |
| usedToStudy |
complex dynamical systems
ⓘ
holomorphic mappings between complex manifolds ⓘ hyperbolicity properties of complex manifolds ⓘ |
| vanishesIdenticallyOn |
complex Euclidean space C^n
ⓘ
complex projective space ⓘ |
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: Kobayashi metric Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kobayashi metric