The Nash embedding theorem in differential geometry states that every Riemannian manifold can be isometrically embedded in an Euclidean space Rn. Intuitively, this 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 bigger than the dimension of the manifold (roughly the third power of the dimension). The theorem was published in 1965 by John Nash.
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 has maximal rank (the rank being equal to the dimension of M). Furthermore, the map dfp 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.
The proof of the theorem relies on Nash's far-reaching generalization of the implicit function theorem.
References:
- John Nash: "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1965), pp 20-63.