"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 C^{k}, 1 ≤ *k* ≤ ∞), then there exists a number *n* and an injective map *f* : *M* `->` **R**^{n} (also analytic or of class C^{k}) such that for every point *p* of *M*, the derivative d*f*_{p} is a linear map from the tangent space T_{p}*M* to **R**^{n} which is compatible with the given inner product on T_{p}*M* and the standard dot product of **R**^{n} in the following sense:

- <
*u*,*v*> = d*f*_{p}(*u*) · d*f*_{p}(*v*)

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into **R**^{n}. 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).

- John Nash: "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1956), pp 20-63.