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 (C^{1}) embeddings and the second for analytic embeddings or embeddings that are smooth of class C^{k}, 3 ≤ k ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and is very counterintuitive, while the proof of the second one is very technical but the result is not at all surprising.
The C^{1} theorem was published in 1954, the C^{k}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 Elie 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 C^{k} case was later extrapolated into the hprinciple 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–Kuiper theorem (C^{1} embedding theorem)
Theorem. Let (M,g) be a Riemannian manifold and ƒ:M^{m} → R^{n} a short C^{∞}embedding (or immersion) into Euclidean space R^{n}, where n ≥ m+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ_{ε} : M^{m} → R^{n} which is
In particular, as follows from the Whitney embedding theorem, any mdimensional Riemannian manifold admits an isometric C^{1}embedding into an arbitrarily small neighborhood in 2mdimensional Euclidean space.
The theorem was originally proved by John Nash with the condition n ≥ m+2 instead of n ≥ m+1 and generalized by Nicolaas Kuiper, by a relatively easy trick.
The theorem has many counterintuitive implications. For example, it follows that any closed oriented Riemannian surface can be C^{1} isometrically embedded into an arbitrarily small ball in Euclidean 3space (from the Gauss–Bonnet theorem, there is no such C^{2}embedding). And, there exist C^{1} isometric embeddings of the hyperbolic plane in R^{3}.
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