"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:
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.
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