Nash embedding theorems

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor 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 embeddings that are analytic or smooth of class C^k, 3 ≤ k ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result.

The C¹ 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 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 C^k- case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simpler 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 an m-dimensional Riemannian manifold and ƒ: M → Rⁿ a short C^∞-embedding (or immersion) into Euclidean space Rⁿ, where n ≥ m+1. This embedding is not required to be isometric. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ_ε: M → Rⁿ which is

1. in class C¹,
2. isometric: for any two vectors v,w ∈ T_x(M) in the tangent space at x ∈ M,

   $g(v,w)=\langle df_{\epsilon }(v),df_{\epsilon }(w)\rangle$,

3. ε-close to ƒ:
   $|f(x)-f_{\epsilon }(x)|<\epsilon ~\forall ~x\in M$.

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. His method was modified by Nicolaas Kuiper to allow n = m + 1.

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 $\epsilon$ 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.

C^k embedding theorem

The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C^k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2, if M is a compact manifold n ≤ m(m+1), or (3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → Rⁿ (also analytic or of class C^k). That is ƒ is an embedding of C^k manifolds and for every point p of M, the derivative dƒ_p is a linear map from the tangent space T_pM to Rⁿ which is compatible with the given inner product on T_pM and the standard dot product of Rⁿ in the following sense:

$\langle u,v\rangle =df_{p}(u)\cdot df_{p}(v)$

for all vectors u, v in T_pM. 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 in a coordinate neighborhood of the manifold. The proof of the global embedding theorem relies on Nash's implicit function theorem for isometric embeddings. This theorem has been generalized by a number of other authors to abstract contexts, where it is known as Nash–Moser theorem. The basic idea in the proof of Nash's implicit function theorem is the use of Newton's method to prove the existence of a solutions. 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).

References

• Eliashberg, Y.; Mishachev, N. (2002). Introduction to the h-principle. Graduate Studies in Mathematics. Vol. 48. Providence, RI: American Mathematical Society. doi:10.1090/gsm/048. ISBN 0-8218-3227-1. MR 1909245.
• Greene, Robert E.; Jacobowitz, Howard (1971). "Analytic isometric embeddings". Annals of Mathematics. Second Series. 93 (1): 189–204. doi:10.2307/1970760. JSTOR 1970760. MR 0283728.
• Gromov, Mikhael (1986). Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 9. Berlin: Springer-Verlag. doi:10.1007/978-3-662-02267-2. ISBN 3-540-12177-3. MR 0864505.
• Günther, Matthias (1989). "Zum Einbettungssatz von J. Nash" [On the embedding theorem of J. Nash]. Mathematische Nachrichten (in German). 144 (1): 165–187. doi:10.1002/mana.19891440113. MR 1037168.
• Kuiper, Nicolaas H. (1955a). "On C¹-isometric imbeddings. I". Indagationes Mathematicae (Proceedings). 58: 545–556. doi:10.1016/S1385-7258(55)50075-8. MR 0075640.
• Kuiper, Nicolaas H. (1955b). "On C¹-isometric imbeddings. II". Indagationes Mathematicae (Proceedings). 58: 683–689. doi:10.1016/S1385-7258(55)50093-X. MR 0075640.
• Nash, John (1954). "C¹ isometric imbeddings". Annals of Mathematics. Second Series. 60 (3): 383–396. doi:10.2307/1969840. JSTOR 1969840. MR 0065993.
• Nash, John (1956). "The imbedding problem for Riemannian manifolds". Annals of Mathematics. Second Series. 63 (1): 20–63. doi:10.2307/1969989. JSTOR 1969989. MR 0075639.
• Nash, J. (1966). "Analyticity of the solutions of implicit function problem with analytic data". Annals of Mathematics. Second Series. 84 (3): 345–355. doi:10.2307/1970448. JSTOR 1970448. MR 0205266.