Nash embedding theorem

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The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn.

"Isometrically" means "preserving lengths of curves". The result therefore means that any Riemannian manifold can be visualized as a submanifold of Euclidean space.

The first theorem is for C1-smooth embeddings and the second for analytic or of class Ck, 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 C1 theorem was published in 1954, the Ck-theorem in 1956, and the analytical case was done in 1966 by John Nash. See h-principle for further developments.

[edit] Nash-Kuiper theorem (C1 embedding theorem)

Theorem. Let (M,g) be a Riemannian manifold and f:M^m\to E^n a short C∞ smooth embedding (or immersion) into Euclidean space En, n\ge m+1. Then for arbitrary ε > 0 there is an embedding (or immersion) f_\epsilon:M^m\to E^n which is

(i) C1-smooth,
(ii) isometric, i.e. for any two vectors v,w\in T_x(M) in the tangent space at x ∈ M we have that g(v,w)=\langle df_\epsilon(v),df_\epsilon(w)\rangle.
(iii) ε-close to f, i.e. : | f(x) − fε(x) | < ε for any x\in M.

In particular, as it follows from Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C1-embedding into an arbitrarily small neighborhood of 2m-dimensional Euclidean space.

The theorem was originally proved by J. Nash with condition n\ge m+2 instead of n\ge 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 ball in Euclidean 3-space (from Gauss formula, there is no such C2-embedding).

[edit] 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 (n = m2+5m+3 will do) and an injective map f : M → Rn (also analytic or of class Ck) such that for every point p of M, the derivative dfp 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:

〈 u, v 〉 = dfp(u) · dfp(v)

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 (see ref.). 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 Kantovorich iteration that uses Newton's method directly (without the introduction of smoothing operators).

[edit] References

  • N.H.Kuiper: "On C1-isometric imbeddings I", Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), pp 545-556.
  • John Nash: "C1-isometric imbeddings", Annals of Mathematics, 60 (1954), pp 383-396.
  • John Nash: "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1956), pp 20-63.
  • John Nash: "Analyticity of the solutions of implicit function problem with analytic data" Annals of Mathematics, 84 (1966), pp 345-355.
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