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.
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi NERFINISHED ⓘ |
| reducesTo | Poisson structure when E = 0 ⓘ |
| relatedConcept |
Lie algebroid
NERFINISHED
ⓘ
Poisson manifold NERFINISHED ⓘ Schouten–Nijenhuis bracket NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.