Hodge star operator
E876575
The Hodge star operator is a linear map on differential forms in oriented Riemannian manifolds that sends a k-form to an (n−k)-form in a way that encodes the metric and orientation, enabling duality operations such as defining codifferentials and expressing vector calculus identities.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hodge star operator canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10661695 — 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: Hodge star operator Context triple: [Levi-Civita symbol, relatedConcept, Hodge star operator]
-
A.
Hodge Laplacian
The Hodge Laplacian is a differential operator on differential forms of a Riemannian manifold that combines the exterior derivative and its adjoint to study harmonic forms and de Rham cohomology.
-
B.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
C.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
-
D.
Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
-
E.
Dirac operator
The Dirac operator is a fundamental first-order differential operator on spinor fields that generalizes the classical Dirac equation and plays a central role in geometry, topology, and quantum field theory.
- 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: Hodge star operator Target entity description: The Hodge star operator is a linear map on differential forms in oriented Riemannian manifolds that sends a k-form to an (n−k)-form in a way that encodes the metric and orientation, enabling duality operations such as defining codifferentials and expressing vector calculus identities.
-
A.
Hodge Laplacian
The Hodge Laplacian is a differential operator on differential forms of a Riemannian manifold that combines the exterior derivative and its adjoint to study harmonic forms and de Rham cohomology.
-
B.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
C.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
-
D.
Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
-
E.
Dirac operator
The Dirac operator is a fundamental first-order differential operator on spinor fields that generalizes the classical Dirac equation and plays a central role in geometry, topology, and quantum field theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
duality operator
ⓘ
linear operator ⓘ mathematical operator ⓘ |
| actsOn |
differential forms
ⓘ
exterior algebra of the cotangent bundle ⓘ |
| appearsIn |
Maxwell's equations in differential form notation
ⓘ
gauge theory ⓘ general relativity formulations using differential forms ⓘ |
| codomain | (n−k)-forms on an oriented Riemannian manifold ⓘ |
| context |
oriented Riemannian manifolds
ⓘ
pseudo-Riemannian manifolds ⓘ |
| definedBy | requirement that α ∧ *β equals the inner product of α and β times the volume form ⓘ |
| dependsOn |
dimension of the manifold
ⓘ
signature of the metric ⓘ |
| domain | k-forms on an oriented Riemannian manifold ⓘ |
| encodes |
metric information
ⓘ
orientation information ⓘ |
| field |
Riemannian geometry
ⓘ
linked to:
Riemannian manifolds
differential geometry ⓘ geometric analysis ⓘ |
| generalizationOf |
cross product in three dimensions via forms
ⓘ
orthogonal complement operation in inner product spaces ⓘ |
| importantIn |
Yang–Mills theory
ⓘ
elliptic partial differential equations on manifolds ⓘ the study of harmonic forms ⓘ |
| is |
linear over the complex numbers
ⓘ
linear over the real numbers ⓘ |
| namedAfter | W. V. D. Hodge ⓘ |
| notation | * ⓘ |
| property |
is an isometry up to sign with respect to the induced inner product on forms
ⓘ
maps volume form to 1 and 1 to volume form up to sign ⓘ |
| relatedTo |
Hodge theory
ⓘ
Laplace–de Rham operator ⓘ
linked to:
Hodge Laplacian
Poincaré duality ⓘ codifferential ⓘ exterior derivative ⓘ |
| requiresStructure |
Riemannian metric
ⓘ
linked to:
Riemannian manifolds
orientation ⓘ |
| satisfies |
* * α = (−1)^{k(n−k)} α on k-forms in n dimensions
ⓘ
is an involution up to sign ⓘ is an isomorphism between k-forms and (n−k)-forms ⓘ |
| usedFor |
defining inner products on differential forms
ⓘ
defining self-dual and anti-self-dual forms ⓘ defining the Laplace–de Rham operator ⓘ defining the codifferential ⓘ expressing vector calculus identities ⓘ formulating Hodge decomposition ⓘ |
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: Hodge star operator Description of subject: The Hodge star operator is a linear map on differential forms in oriented Riemannian manifolds that sends a k-form to an (n−k)-form in a way that encodes the metric and orientation, enabling duality operations such as defining codifferentials and expressing vector calculus identities.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.