Jacobi manifold
E697764
differential-geometric structure
geometric structure
mathematical concept
smooth manifold with additional structure
A Jacobi manifold is a smooth manifold equipped with a Lie bracket on its space of smooth functions that satisfies a generalized Leibniz rule, extending the notion of Poisson manifolds.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi manifold canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871937 — 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: Jacobi manifold Context triple: [Jacobi bracket, relatedTo, Jacobi manifold]
-
A.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
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B.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
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C.
Lie algebroid
A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
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D.
Jacobi last multiplier
The Jacobi last multiplier is a mathematical tool introduced by Carl Gustav Jacob Jacobi for integrating systems of differential equations by providing an integrating factor that simplifies them to solvable form.
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E.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
- 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: Jacobi manifold Target entity description: A Jacobi manifold is a smooth manifold equipped with a Lie bracket on its space of smooth functions that satisfies a generalized Leibniz rule, extending the notion of Poisson manifolds.
-
A.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
-
B.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
-
C.
Lie algebroid
A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
-
D.
Jacobi last multiplier
The Jacobi last multiplier is a mathematical tool introduced by Carl Gustav Jacob Jacobi for integrating systems of differential equations by providing an integrating factor that simplifies them to solvable form.
-
E.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
differential-geometric structure
ⓘ
geometric structure ⓘ mathematical concept ⓘ smooth manifold with additional structure ⓘ |
| allows | non-homogeneous Hamiltonian dynamics ⓘ |
| associatedTo |
contact manifold
ⓘ
locally conformal symplectic manifold ⓘ |
| bracketFormula | {f,g} = Λ(df,dg) + fE(g) − gE(f) ⓘ |
| bracketNotation | {f,g} for smooth functions f,g ⓘ |
| canBeDescribedBy | pair (Λ,E) where Λ is a bivector field and E is a vector field ⓘ |
| conditionOnPair |
[E,Λ] = 0 (Schouten–Nijenhuis bracket)
ⓘ
[Λ,Λ] = 2E ∧ Λ (Schouten–Nijenhuis bracket) ⓘ |
| definedOn | space of smooth real-valued functions on the manifold ⓘ |
| equippedWith |
Lie bracket on smooth functions
ⓘ
local Lie algebra structure on C^\\infty(M) ⓘ |
| field |
Poisson geometry
ⓘ
differential geometry ⓘ mathematical physics ⓘ symplectic geometry ⓘ |
| generalizes |
Poisson manifold
ⓘ
local Poisson brackets on functions ⓘ |
| hasAssociatedAlgebroid | Lie algebroid on T^*M ⊕ ℝ ⓘ |
| hasLocalModel |
contact structure on odd-dimensional manifolds
ⓘ
locally conformal symplectic structure on even-dimensional manifolds ⓘ |
| hasMorphisms | Jacobi maps ⓘ |
| hasStructure | bivector field and vector field description ⓘ |
| hasUnderlyingObject | smooth manifold ⓘ |
| inducedBy |
contact structure on a manifold
ⓘ
locally conformal symplectic structure ⓘ |
| introducedBy | André Lichnerowicz ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| reducesTo | Poisson structure when E = 0 ⓘ |
| relatedConcept |
Lie algebroid
ⓘ
Poisson manifold ⓘ Schouten–Nijenhuis bracket ⓘ contact manifold ⓘ symplectic manifold ⓘ |
| satisfiesProperty |
Jacobi identity for the bracket
ⓘ
antisymmetry of the bracket ⓘ bilinearity of the bracket ⓘ generalized Leibniz rule ⓘ |
| specialCaseOf | local Lie algebra ⓘ |
| usedIn |
classical mechanics
ⓘ
geometric quantization ⓘ theory of constraints in Hamiltonian systems ⓘ |
| yearIntroducedApprox | 1978 ⓘ |
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: Jacobi manifold Description of subject: A Jacobi manifold is a smooth manifold equipped with a Lie bracket on its space of smooth functions that satisfies a generalized Leibniz rule, extending the notion of Poisson manifolds.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.