Nash embedding theorem

"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 -> Rⁿ (also analytic or of class C^k) such that for every point p of M, the derivative df_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 > = df_p(u) · df_p(v)

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.

References

• John Nash: "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1965), pp 20-63.