Kähler manifold
E23190
A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Kähler manifold canonical | 3 |
| Kähler geometry | 2 |
| Kähler manifolds | 1 |
| Kähler metric | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T179366 — 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: Kähler manifold Context triple: [Riemannian manifold, hasVariant, Kähler manifold]
-
A.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
-
B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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C.
Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
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D.
Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
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E.
Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
- 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: Kähler manifold Target entity description: A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
-
A.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
-
B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
-
D.
Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
-
E.
Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Riemannian manifold
ⓘ
complex manifold ⓘ geometric structure ⓘ symplectic manifold ⓘ |
| associatedForm |
Kähler form
ⓘ
fundamental 2-form ⓘ |
| definitionCondition |
J is orthogonal with respect to the Riemannian metric
ⓘ
Kähler form is d-closed ⓘ ∇J = 0 for Levi-Civita connection ∇ ⓘ |
| dimension |
complex dimension n
ⓘ
even real dimension ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ differential geometry ⓘ symplectic geometry ⓘ |
| hasCohomologyProperty |
Betti numbers satisfy b_{2k+1} is even
ⓘ
admits Hodge decomposition ⓘ satisfies Hodge symmetry ⓘ satisfies hard Lefschetz theorem ⓘ |
| hasCurvatureProperty | Riemann curvature tensor has Kähler symmetries ⓘ |
| hasExample |
Calabi–Yau manifold
ⓘ
Riemann surface with any Hermitian metric ⓘ complex projective space CP^n with Fubini–Study metric ⓘ complex tori with flat metric ⓘ smooth projective algebraic variety over C ⓘ |
| hasOperator |
Dolbeault operators ∂ and ∂̄
ⓘ
Lefschetz operator ⓘ |
| hasProperty |
Kähler form is closed
ⓘ
Kähler identities hold between ∂, ∂̄, and Lefschetz operators ⓘ Laplace–Beltrami operator equals Hodge Laplacian on forms ⓘ Levi-Civita connection preserves complex structure ⓘ complex structure is integrable ⓘ holonomy group is contained in U(n) ⓘ metric is Hermitian with respect to complex structure ⓘ symplectic form is compatible with complex structure ⓘ |
| hasStructure |
Hermitian metric
ⓘ
Riemannian metric ⓘ complex structure ⓘ symplectic form ⓘ |
| implies |
underlying manifold is Riemannian
ⓘ
underlying manifold is complex ⓘ underlying manifold is symplectic ⓘ |
| localCoordinateDescription |
Kähler form is i∂∂̄ of a real-valued potential function (locally)
ⓘ
metric is given by a Kähler potential ⓘ |
| namedAfter | Erich Kähler ⓘ |
| usedIn |
Hodge theory
ⓘ
algebraic geometry ⓘ string 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: Kähler manifold Description of subject: A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Riemannian manifold
this entity surface form:
Kähler geometry
this entity surface form:
Kähler geometry
this entity surface form:
Kähler manifolds
this entity surface form:
Kähler metric