Nash embedding theorem
E631
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.
Aliases (8)
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 |
general relativity
→
the study of manifolds with given metric structures → |
| hasProperty |
global embedding result
→
nonlinear partial differential equation method → |
| hasVersion |
Nash C1 embedding theorem
→
Nash Ck embedding theorem → Nash C∞ embedding theorem → Nash isometric embedding theorem → 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 Forbes Nash Jr.
→
|
| provedBy |
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
→
|
Referenced by (10)
| Subject (surface form when different) | Predicate |
|---|---|
|
Nash embedding theorem
("Nash C1 embedding theorem")
→
Nash embedding theorem ("Nash Ck embedding theorem") → Nash embedding theorem ("Nash C∞ embedding theorem") → Nash embedding theorem ("Nash isometric embedding theorem") → Nash embedding theorem ("Nash–Kuiper theorem") → |
hasVersion |
|
Janet–Cartan theorem
("Nash embedding theorems")
→
Whitney embedding theorem → |
relatedTo |
|
Janet–Cartan theorem
("Nash C^k isometric embedding theorem")
→
Janet–Cartan theorem ("Nash C^1 isometric embedding theorem") → |
strengthenedBy |
|
John Nash
→
|
notableWork |