Nash embedding theorem

E631

mathematical theorem result in differential geometry

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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All labels observed (9)

Label	Occurrences
Nash embedding theorem canonical	2
Nash C1 embedding theorem	1
Nash C^1 isometric embedding theorem	1
Nash C^k isometric embedding theorem	1
Nash Ck embedding theorem	1
Nash C∞ embedding theorem	1
Nash embedding theorems	1
Nash isometric embedding theorem	1
Nash–Kuiper theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in differential geometry ⓘ
appliesTo	Ck Riemannian metrics ⓘ compact Riemannian manifolds ⓘ noncompact Riemannian manifolds ⓘ smooth Riemannian manifolds ⓘ
clarifies	relationship between intrinsic curvature and extrinsic curvature ⓘ
concerns	Euclidean space ⓘ Riemannian manifolds ⓘ isometric embeddings ⓘ
dimensionBound	gives explicit upper bounds on the Euclidean dimension needed for embedding ⓘ
field	Riemannian geometry ⓘ differential geometry ⓘ
generalizationOf	local isometric embedding results ⓘ
hasConsequence	Riemannian manifolds can be studied via submanifolds of Euclidean space ⓘ any Riemannian manifold isometrically embeds into some RN ⓘ existence of isometric embeddings for compact Riemannian manifolds ⓘ existence of isometric embeddings for noncompact Riemannian manifolds ⓘ
hasImpactOn	theory of relativity ⓘ surface form: general relativity the study of manifolds with given metric structures ⓘ
hasProperty	global embedding result ⓘ nonlinear partial differential equation method ⓘ
hasVersion	Nash embedding theorem self-linksurface differs ⓘ surface form: Nash C1 embedding theorem Nash embedding theorem self-linksurface differs ⓘ surface form: Nash Ck embedding theorem Nash embedding theorem self-linksurface differs ⓘ surface form: Nash C∞ embedding theorem Nash embedding theorem self-linksurface differs ⓘ surface form: Nash isometric embedding theorem Nash embedding theorem self-linksurface differs ⓘ surface form: Nash–Kuiper theorem
implies	every abstract Riemannian manifold can be realized as a submanifold of Euclidean space ⓘ
influenced	geometric analysis ⓘ global Riemannian geometry ⓘ theory of isometric immersions ⓘ
isStrongerThan	local isometric embedding theorems ⓘ
namedAfter	John Nash ⓘ surface form: John Forbes Nash Jr.
provedBy	John Nash ⓘ surface form: John Forbes Nash Jr.
relatedTo	Janet–Cartan theorem ⓘ Whitney embedding theorem ⓘ
relatesConcept	embedding ⓘ extrinsic geometry ⓘ immersion ⓘ intrinsic geometry ⓘ isometry ⓘ metric tensor ⓘ
shows	intrinsic Riemannian geometry can be realized as extrinsic geometry in Euclidean space ⓘ
statesThat	every smooth Riemannian manifold admits an isometric embedding into some Euclidean space ⓘ
usesMethod	implicit function theorem ⓘ iteration scheme ⓘ smoothing operators ⓘ
yearProved	1950s ⓘ