Poisson geometry
E697763
Poisson geometry is the branch of differential geometry that studies manifolds equipped with a Poisson bracket, generalizing classical Hamiltonian mechanics and symplectic geometry.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Poisson geometry canonical | 1 |
| Poisson manifold | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871924 — 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: Poisson geometry Context triple: [Jacobi bracket, field, Poisson geometry]
-
A.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
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B.
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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C.
Lie sphere geometry
Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.
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D.
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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E.
differential geometry
Differential geometry is a branch of mathematics that uses the techniques of calculus and linear algebra to study the properties and curvature of smooth shapes and spaces such as curves, surfaces, and manifolds.
- 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: Poisson geometry Target entity description: Poisson geometry is the branch of differential geometry that studies manifolds equipped with a Poisson bracket, generalizing classical Hamiltonian mechanics and symplectic geometry.
-
A.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
-
B.
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.
-
C.
Lie sphere geometry
Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.
-
D.
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.
-
E.
differential geometry
Differential geometry is a branch of mathematics that uses the techniques of calculus and linear algebra to study the properties and curvature of smooth shapes and spaces such as curves, surfaces, and manifolds.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
branch of differential geometry
ⓘ
mathematical discipline ⓘ |
| characterizedBy |
Jacobi identity
ⓘ
Leibniz rule NERFINISHED ⓘ bilinear Poisson bracket ⓘ skew-symmetric bracket ⓘ |
| developedIn | 20th century ⓘ |
| fieldOfStudy |
Hamiltonian systems
ⓘ
Poisson brackets ⓘ Poisson manifolds ⓘ symplectic geometry ⓘ |
| formalizedBy |
Poisson algebra
NERFINISHED
ⓘ
Poisson manifold NERFINISHED ⓘ |
| generalizes |
classical Hamiltonian mechanics
ⓘ
symplectic geometry ⓘ |
| hasApplicationIn |
classical mechanics
ⓘ
field theory ⓘ quantization theory ⓘ representation theory ⓘ |
| hasNotableContributor |
Alan Weinstein
NERFINISHED
ⓘ
André Lichnerowicz NERFINISHED ⓘ Jean-Marie Souriau NERFINISHED ⓘ Mikhail Gromov NERFINISHED ⓘ Victor Ginzburg NERFINISHED ⓘ |
| namedAfter | Siméon Denis Poisson NERFINISHED ⓘ |
| relatedTo |
Lie theory
NERFINISHED
ⓘ
deformation quantization ⓘ integrable systems ⓘ mathematical physics ⓘ noncommutative geometry NERFINISHED ⓘ symplectic geometry ⓘ |
| studies |
Lie algebroids
ⓘ
Poisson bivector fields ⓘ coisotropic submanifolds ⓘ integrable systems ⓘ manifolds with Poisson structure ⓘ momentum maps ⓘ symplectic groupoids ⓘ |
| usesConcept |
Casimir function
ⓘ
Hamiltonian vector field ⓘ Lie algebra NERFINISHED ⓘ Lie algebroid ⓘ Lie groupoid NERFINISHED ⓘ Poisson bracket ⓘ Schouten–Nijenhuis bracket NERFINISHED ⓘ bivector field ⓘ cohomology ⓘ foliation ⓘ symplectic form ⓘ |
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: Poisson geometry Description of subject: Poisson geometry is the branch of differential geometry that studies manifolds equipped with a Poisson bracket, generalizing classical Hamiltonian mechanics and symplectic geometry.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poisson manifold