Whitney embedding theorem
E9682
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
All labels observed (3)
| Label | Occurrences |
|---|---|
| Whitney embedding theorem canonical | 7 |
| Whitney immersion theorem | 1 |
| strong Whitney embedding theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T31662 — 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: Whitney embedding theorem Context triple: [Nash embedding theorem, relatedTo, Whitney embedding theorem]
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A.
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.
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B.
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.
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C.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
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D.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
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E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- 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: Whitney embedding theorem
Target entity description: The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
A.
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.
-
B.
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.
-
C.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
-
D.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
-
E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in differential topology ⓘ |
| appliesTo | finite-dimensional smooth manifolds ⓘ |
| category | C^{} (smooth) category ⓘ |
| concerns |
Euclidean space
ⓘ
embeddings ⓘ immersions ⓘ smooth manifolds ⓘ |
| dimensionBound |
minimal embedding dimension is at most 2n
ⓘ
minimal immersion dimension is at most 2n-1 ⓘ |
| field |
differential geometry
ⓘ
differential topology ⓘ topology ⓘ |
| generalizes |
the fact that smooth curves embed in R^3
ⓘ
the fact that smooth surfaces embed in R^5 ⓘ |
| givesUpperBound |
2n for the embedding dimension of an n-dimensional smooth manifold
ⓘ
2n-1 for the immersion dimension of an n-dimensional smooth manifold ⓘ |
| hasConsequence |
existence of smooth embeddings into Euclidean space for compact manifolds
ⓘ
finite-dimensional smooth manifolds can be studied via subsets of Euclidean space ⓘ |
| hasRefinement |
strong Whitney embedding theorem
ⓘ
linked to:
Whitney embedding theorem
|
| hasVariant |
Whitney immersion theorem
ⓘ
linked to:
Whitney embedding theorem
|
| holdsFor |
compact smooth manifolds
ⓘ
non-compact smooth manifolds ⓘ |
| implies |
every smooth manifold is diffeomorphic to a submanifold of some R^N
ⓘ
every smooth n-dimensional manifold can be realized as a submanifold of some Euclidean space ⓘ |
| isFundamentalResultIn | classification of smooth manifolds up to embedding ⓘ |
| isUsedIn |
cobordism theory
ⓘ
construction of smooth structures on manifolds ⓘ geometric topology ⓘ singularity theory ⓘ surgery theory ⓘ |
| namedAfter | Hassler Whitney ⓘ |
| provedBy | Hassler Whitney ⓘ |
| publishedIn | Annals of Mathematics ⓘ |
| relatedTo |
Nash embedding theorem
ⓘ
Whitney approximation theorem ⓘ Whitney stratification ⓘ |
| requires |
Hausdorff manifold
ⓘ
second countable manifold ⓘ |
| states |
every smooth n-dimensional manifold admits an embedding into R^{2n}
ⓘ
every smooth n-dimensional manifold admits an immersion into R^{2n-1} ⓘ |
| usesConcept |
approximation by embeddings
ⓘ
differentiable map ⓘ injective immersion ⓘ smooth structure ⓘ submanifold ⓘ topological embedding ⓘ transversality ⓘ |
| yearProved | 1944 ⓘ |
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: Whitney embedding theorem
Description of subject: The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Whitney embedding theorem
linked to: Whitney embedding theorem
subject linked to:
Hassler
subject linked to:
M. Hirsch, Differential Topology