Nash embedding theorem
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–Kuiper theorem (C1 embedding theorem) 
Theorem. Let (M,g) be a Riemannian manifold and ƒ: Mm → Rn a short C∞-embedding (or immersion) into Euclidean space Rn, where n ≥ m+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒε: Mm → Rn which is
- (i) in class C1,
- (ii) isometric: for any two vectors v,w ∈ Tx(M) in the tangent space at x ∈ M,
- (iii) ε-close to ƒ:
In particular, as follows from the Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C1-embedding into an arbitrarily small neighborhood in 2m-dimensional 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 C1 isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space (for small there is no such C2-embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε-2). And, there exist C1 isometric embeddings of the hyperbolic plane in R3.
Ck embedding theorem
The technical statement is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class Ck, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, or n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an injective map ƒ: M → Rn (also analytic or of class Ck) such that for every point p of M, the derivative dƒp is a linear map from the tangent space TpM to Rn which is compatible with the given inner product on TpM and the standard dot product of Rn in the following sense:
for all vectors u, v in TpM. This is an undetermined system of partial differential equations (PDEs).
The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into Rn. A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus. The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the Nash–Moser theorem and Newton's method with postconditioning. The basic idea of Nash's solution of the embedding problem is the use of Newton's method to prove the existence of a solution to the above system of PDEs. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by convolution to make the Newton iteration converge: this is Newton's method with postconditioning. The fact that this technique furnishes a solution is in itself an existence theorem and of independent interest. There is also an older method called Kantorovich iteration that uses Newton's method directly (without the introduction of smoothing operators).
- Greene, Robert E.; Jacobowitz, Howard (1971), "Analytic Isometric Embeddings", Annals of Mathematics 93 (1): 189–204, doi:10.2307/1970760, JSTOR 1970760
- Günther, Matthias (1989), "Zum Einbettungssatz von J. Nash [On the embedding theorem of J. Nash]", Mathematische Nachrichten 144: 165–187, doi:10.1002/mana.19891440113
- Han, Qing; Hong, Jia-Xing (2006), Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, American Mathematical Society, ISBN 0-8218-4071-1
- Kuiper, N.H. (1955), "On C1-isometric imbeddings I", Nederl. Akad. Wetensch. Proc. Ser. A. 58: 545–556.
- Nash, John (1954), "C1-isometric imbeddings", Annals of Mathematics 60 (3): 383–396, doi:10.2307/1969840, JSTOR 1969840.
- Nash, John (1956), "The imbedding problem for Riemannian manifolds", Annals of Mathematics 63 (1): 20–63, doi:10.2307/1969989, JSTOR 1969989, MR 0075639.
- Nash, John (1966), "Analyticity of the solutions of implicit function problem with analytic data", Annals of Mathematics 84 (3): 345–355, doi:10.2307/1970448, JSTOR 1970448.