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- Nash_embedding_theorem abstract "The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent.The first theorem is for continuously differentiable (C1) embeddings and the second for analytic embeddings or embeddings that are smooth of class Ck, 3 ≤ k ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and leads to some very counterintuitive conclusions, while the proof of the second one is very technical but the result is not that surprising.The C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by Greene & Jacobowitz (1971). (A local version of this result was proved by Élie Cartan and Maurice Janet in the 1920s.) In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the Ck- case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simplified proof of the second Nash embedding theorem was obtained by Günther (1989) who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied.".
- Nash_embedding_theorem wikiPageExternalLink Erratum.txt.
- Nash_embedding_theorem wikiPageID "51129".
- Nash_embedding_theorem wikiPageRevisionID "599791485".
- Nash_embedding_theorem hasPhotoCollection Nash_embedding_theorem.
- Nash_embedding_theorem subject Category:Theorems_in_Riemannian_geometry.
- Nash_embedding_theorem type Abstraction100002137.
- Nash_embedding_theorem type Communication100033020.
- Nash_embedding_theorem type Message106598915.
- Nash_embedding_theorem type Proposition106750804.
- Nash_embedding_theorem type Statement106722453.
- Nash_embedding_theorem type Theorem106752293.
- Nash_embedding_theorem type TheoremsInRiemannianGeometry.
- Nash_embedding_theorem comment "The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path.".
- Nash_embedding_theorem label "Einbettungssatz von Nash".
- Nash_embedding_theorem label "Inbeddingstelling van Nash".
- Nash_embedding_theorem label "Nash embedding theorem".
- Nash_embedding_theorem label "Teorema de imersão de Nash".
- Nash_embedding_theorem label "Teorema de inmersión de Nash".
- Nash_embedding_theorem label "Théorème de plongement de Nash".
- Nash_embedding_theorem label "Теорема Нэша о регулярных вложениях".
- Nash_embedding_theorem label "纳什嵌入定理".
- Nash_embedding_theorem sameAs Einbettungssatz_von_Nash.
- Nash_embedding_theorem sameAs Teorema_de_inmersión_de_Nash.
- Nash_embedding_theorem sameAs Théorème_de_plongement_de_Nash.
- Nash_embedding_theorem sameAs Inbeddingstelling_van_Nash.
- Nash_embedding_theorem sameAs Teorema_de_imersão_de_Nash.
- Nash_embedding_theorem sameAs m.0dh6t.
- Nash_embedding_theorem sameAs Q1306092.
- Nash_embedding_theorem sameAs Q1306092.
- Nash_embedding_theorem sameAs Nash_embedding_theorem.
- Nash_embedding_theorem wasDerivedFrom Nash_embedding_theorem?oldid=599791485.
- Nash_embedding_theorem isPrimaryTopicOf Nash_embedding_theorem.