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Nash embedding theorem

The Nash embedding theorem (or imbedding theorem) in differential geometry, published in 1956 by John Nash, states that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn.

"Isometrically" means "preserving distances and angles" (the part about angles is actually redundant; if a mapping preserves distances then a fortiori it must preserve angles). Intuitively, the result therefore means that the notion of length and angle given on a Riemannian manifold can be visualized as the familiar notions of length and angle in Euclidean space. Note however that the number n is in general much larger than the dimension of the manifold (roughly the third power of the dimension).

The technical statement is as follows: if M is a given Riemannian manifold (analytic or of class Ck, 1 ≤ k ≤ ∞), then there exists a number n 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 (PDE's).

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 as presented here relies on Nash's far-reaching generalization of the implicit function theorem, the Nash-Moser inverse function theorem and Newton's method with postconditioning see ref. The basic idea of Nash to solve the embedding problem was to use Newton's method to prove the system of PDE's has a solution. The standard Newton method fails to converge when applied to the system, so Nash uses smoothing operators to ensure to make the Newton iteration converge this adapted Newton method is called Newton method with postconditioning. The smoothing operators are defined by convolution. The smoothing operators ensure that the iteration converges to a root and so it can be used as an existence theorem as well. By showing that the systems of PDE's has a root proves the existence of isometric embedding of Riemannian manifolds. There is also a older iteration called the Kantovorich iteration that is an existence theorem using only Newton's Method (so no smoothing operators).