Nash embedding theorems

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. The theorem was published in 1965 by John Nash.

The technical statement is as follows: if M is a given Riemannian manifold, then there exists a number n and an injective analytical map f : M -> Rⁿ 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 has maximal rank (the rank being equal to the dimension of M). Furthermore, the map df_p 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)

for all vectors u, v in T_pM.

References:

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