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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.".
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- Nash_embedding_theorem wikiPageRevisionID "665139145".
- Nash_embedding_theorem wikiPageWikiLink Analytic_function.
- Nash_embedding_theorem wikiPageWikiLink Annals_of_Mathematics.
- Nash_embedding_theorem wikiPageWikiLink Arc_length.
- Nash_embedding_theorem wikiPageWikiLink Ball_(mathematics).
- Nash_embedding_theorem wikiPageWikiLink Banach_fixed-point_theorem.
- Nash_embedding_theorem wikiPageWikiLink Category:Theorems_in_Riemannian_geometry.
- Nash_embedding_theorem wikiPageWikiLink Convolution.
- Nash_embedding_theorem wikiPageWikiLink Curve.
- Nash_embedding_theorem wikiPageWikiLink Derivative.
- Nash_embedding_theorem wikiPageWikiLink Differentiable_function.
- Nash_embedding_theorem wikiPageWikiLink Dot_product.
- Nash_embedding_theorem wikiPageWikiLink Embedding.
- Nash_embedding_theorem wikiPageWikiLink Euclidean_space.
- Nash_embedding_theorem wikiPageWikiLink Existence_theorem.
- Nash_embedding_theorem wikiPageWikiLink Gaussian_curvature.
- Nash_embedding_theorem wikiPageWikiLink Homotopy_principle.
- Nash_embedding_theorem wikiPageWikiLink Immersion_(mathematics).
- Nash_embedding_theorem wikiPageWikiLink Implicit_function_theorem.
- Nash_embedding_theorem wikiPageWikiLink Injective_function.
- Nash_embedding_theorem wikiPageWikiLink Inner_product_space.
- Nash_embedding_theorem wikiPageWikiLink Isometry.
- Nash_embedding_theorem wikiPageWikiLink John_Forbes_Nash,_Jr..
- Nash_embedding_theorem wikiPageWikiLink Kantorovich_theorem.
- Nash_embedding_theorem wikiPageWikiLink Linear_map.
- Nash_embedding_theorem wikiPageWikiLink Manifold.
- Nash_embedding_theorem wikiPageWikiLink Mathematische_Nachrichten.
- Nash_embedding_theorem wikiPageWikiLink Maurice_Janet.
- Nash_embedding_theorem wikiPageWikiLink Metric_map.
- Nash_embedding_theorem wikiPageWikiLink Nash–Moser_theorem.
- Nash_embedding_theorem wikiPageWikiLink Newtons_method.
- Nash_embedding_theorem wikiPageWikiLink Nicolaas_Kuiper.
- Nash_embedding_theorem wikiPageWikiLink Partial_differential_equation.
- Nash_embedding_theorem wikiPageWikiLink Riemannian_manifold.
- Nash_embedding_theorem wikiPageWikiLink Smoothness.
- Nash_embedding_theorem wikiPageWikiLink Tangent_space.
- Nash_embedding_theorem wikiPageWikiLink Whitney_embedding_theorem.
- Nash_embedding_theorem wikiPageWikiLink Élie_Cartan.
- Nash_embedding_theorem wikiPageWikiLinkText "Nash embedding theorem".
- Nash_embedding_theorem wikiPageWikiLinkText "Nash embedding theorem#Nash–Kuiper theorem".
- Nash_embedding_theorem wikiPageUsesTemplate Template:Anchor.
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- Nash_embedding_theorem wikiPageUsesTemplate Template:Harvtxt.
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- Nash_embedding_theorem subject Category:Theorems_in_Riemannian_geometry.
- Nash_embedding_theorem type Redirect.
- Nash_embedding_theorem type Theorem.
- 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 "Nash embedding theorem".
- Nash_embedding_theorem sameAs Q1306092.
- 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 ナッシュの埋め込み定理.
- 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 Теорема_Нэша_о_регулярных_вложениях.
- Nash_embedding_theorem sameAs Q1306092.
- Nash_embedding_theorem sameAs 纳什嵌入定理.
- Nash_embedding_theorem wasDerivedFrom Nash_embedding_theorem?oldid=665139145.
- Nash_embedding_theorem isPrimaryTopicOf Nash_embedding_theorem.