Whitney embedding theorem

E9682

mathematical theorem theorem in differential topology

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 \(\mathbb{R}^{2n}\)).

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All labels observed (3)

Label	Occurrences
Whitney embedding theorem canonical	6
Whitney immersion theorem	1
strong Whitney embedding theorem	1

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	Whitney embedding theorem self-linksurface differs ⓘ surface form: strong Whitney embedding theorem
hasVariant	Whitney embedding theorem self-linksurface differs ⓘ surface form: Whitney immersion 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 ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Nash embedding theorem → relatedTo → Whitney embedding theorem ⓘ

Whitney embedding theorem → hasRefinement → Whitney embedding theorem self-linksurface differs ⓘ

this entity surface form: strong Whitney embedding theorem

Whitney embedding theorem → hasVariant → Whitney embedding theorem self-linksurface differs ⓘ

this entity surface form: Whitney immersion theorem

Hassler Whitney → notableFor → Whitney embedding theorem ⓘ

Hassler Whitney → hasTheoremNamedAfter → Whitney embedding theorem ⓘ

Whitney approximation theorem → relatedTo → Whitney embedding theorem ⓘ

Hassler → notableFor → Whitney embedding theorem ⓘ

subject surface form: Hassler Whitney

M. Hirsch, Differential Topology → topic → Whitney embedding theorem ⓘ

subject surface form: Differential Topology (book)