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

Aliases (2)

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 →
hasVariant	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 (6)

Subject (surface form when different)	Predicate
Nash embedding theorem → Whitney approximation theorem →	relatedTo
Whitney embedding theorem ("strong Whitney embedding theorem") →	hasRefinement
Hassler Whitney →	hasTheoremNamedAfter
Whitney embedding theorem ("Whitney immersion theorem") →	hasVariant
Hassler Whitney →	notableFor