Darboux theorem
E506852
The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Darboux theorem canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T5256390 — 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: Darboux theorem Context triple: [Carathéodory–Jacobi–Lie theorem, generalizes, Darboux theorem]
-
A.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
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.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
-
E.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
- 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: Darboux theorem Target entity description: The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
-
A.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
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.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
-
E.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo | finite-dimensional symplectic manifolds ⓘ |
| assumes | nondegenerate closed 2-form ⓘ |
| category | results about canonical forms ⓘ |
| concerns |
local structure of symplectic manifolds
ⓘ
symplectic forms ⓘ symplectic manifolds ⓘ |
| conclusion | existence of local coordinates in which the symplectic form is standard ⓘ |
| contrastWith | Riemannian geometry where curvature gives local invariants ⓘ |
| coordinateName | Darboux coordinates ⓘ |
| dimensionCondition | manifold dimension is even ⓘ |
| doesNotDependOn |
choice of Riemannian metric
ⓘ
global topology of the manifold ⓘ |
| domain | smooth manifolds ⓘ |
| field |
differential geometry
ⓘ
symplectic geometry ⓘ |
| formalization | for any point of a symplectic manifold there exist local coordinates making the symplectic form standard ⓘ |
| guaranteesExistenceOf | Darboux coordinates NERFINISHED ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| implies |
every symplectic form admits a canonical local normal form
ⓘ
there are no local invariants of symplectic manifolds other than dimension ⓘ |
| influenceOn |
mathematical formulation of classical mechanics
ⓘ
modern symplectic topology ⓘ |
| involves |
diffeomorphisms preserving the symplectic form
ⓘ
symplectomorphisms ⓘ |
| locality | purely local result ⓘ |
| localModel | standard symplectic vector space (R^{2n}, sum dq_i wedge dp_i) ⓘ |
| namedAfter | Jean Gaston Darboux NERFINISHED ⓘ |
| relatedTo |
Moser trick
NERFINISHED
ⓘ
Weinstein neighborhood theorem NERFINISHED ⓘ canonical coordinates in Hamiltonian mechanics ⓘ normal form theorems in differential geometry ⓘ |
| requires |
closedness of the symplectic form
ⓘ
nondegeneracy of the symplectic form ⓘ |
| shows | symplectic geometry has no local curvature-type invariants ⓘ |
| standardForm | sum of dq_i wedge dp_i in suitable local coordinates ⓘ |
| statement | all symplectic manifolds are locally symplectomorphic to the standard symplectic space ⓘ |
| symplecticFormLocalExpression | omega = sum_{i=1}^n dq_i wedge dp_i in Darboux coordinates ⓘ |
| teaches | all symplectic manifolds are locally indistinguishable as symplectic spaces ⓘ |
| type |
canonical form result
ⓘ
local normal form theorem NERFINISHED ⓘ |
| usedIn |
Hamiltonian mechanics
NERFINISHED
ⓘ
canonical transformations theory ⓘ classical mechanics ⓘ geometric quantization ⓘ study of Poisson manifolds (via symplectic leaves) ⓘ |
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: Darboux theorem Description of subject: The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.