Does the Nash embedding theorem extend to pseudo-Riemannian manifolds?
More precisely, can any ($\mathcal{C}^k$ for some specified $k$) pseudo-Riemannian manifold be isometrically embedded into some pseudo-Euclidean space? If so, can the dimension of the pseudo-Euclidean space be bounded by some absolute function of the manifold's dimension? More strongly, can the positive and negative indices in the pseudo-Euclidean space be separately bounded by some absolute function of the pseudo-Riemannian manifold's metric signature?