Poincaré lemma
E156193
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Poincaré lemma canonical | 1 |
| Poincaré lemma for currents | 1 |
| Poincaré lemma for distributions | 1 |
| Poincaré lemma with parameters | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358652 — 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: Poincaré lemma Context triple: [Henri Poincaré, notableWork, Poincaré lemma]
-
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.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
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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D.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
-
E.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
- 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: Poincaré lemma Target entity description: 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.
-
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.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
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.
-
D.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
-
E.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in algebraic topology ⓘ result in differential geometry ⓘ |
| appliesTo |
open subsets of Euclidean space
ⓘ
smooth manifolds ⓘ |
| assumes |
domain is contractible
ⓘ
forms are smooth ⓘ |
| category | lemma in mathematics ⓘ |
| concernsDegree | k-forms with k > 0 ⓘ |
| coreStatement |
every closed differential form on a star-shaped domain is exact
ⓘ
on a contractible manifold the de Rham cohomology is trivial in positive degrees ⓘ |
| dealsWith |
closed forms
ⓘ
contractible domains ⓘ de Rham cohomology ⓘ differential forms ⓘ exact forms ⓘ star-shaped domains ⓘ |
| failsIn | domains with nontrivial topology ⓘ |
| field |
algebraic topology
ⓘ
differential geometry ⓘ topology ⓘ |
| hasVariant |
Poincaré lemma
self-linksurface differs
ⓘ
surface form:
Poincaré lemma for currents
Poincaré lemma self-linksurface differs ⓘ
surface form:
Poincaré lemma for distributions
Poincaré lemma self-linksurface differs ⓘ
surface form:
Poincaré lemma with parameters
|
| historicalPeriod | late 19th century mathematics ⓘ |
| holdsIn |
contractible smooth manifolds
ⓘ
star-shaped open subsets of R^n ⓘ |
| implies |
closed forms are locally exact
ⓘ
local triviality of de Rham cohomology ⓘ |
| importance |
connects local and global properties of manifolds
ⓘ
fundamental tool in modern differential geometry ⓘ |
| isLocalVersionOf | triviality of higher de Rham cohomology on contractible spaces ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| relatedTo |
Hodge theory
ⓘ
Mayer–Vietoris sequence in de Rham cohomology ⓘ Stokes' theorem ⓘ
surface form:
Stokes theorem
|
| relatesConcept |
closedness of a form
ⓘ
exactness of a form ⓘ exterior derivative ⓘ homotopy operator ⓘ |
| usedIn |
computation of de Rham cohomology groups
ⓘ
gauge theory ⓘ local normal form arguments in differential geometry ⓘ theory of symplectic manifolds ⓘ |
| usedInProofOf |
Poincaré duality
ⓘ
de Rham cohomology ⓘ
surface form:
de Rham theorem
|
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: Poincaré lemma Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré lemma for distributions
this entity surface form:
Poincaré lemma with parameters
this entity surface form:
Poincaré lemma for currents