Nash Embedding Theorem - Ck Embedding Theorem

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 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:

〈 u, v 〉 = dƒ_p(u) · dƒ_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. 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).