Nash Embedding Theorems for Pseudo-Riemannian Manifolds? Are there analogs of the Nash Embedding Theorems for Pseudo-Riemannian Manifolds?
 A: Not clear where you are headed with your concise question, 
but if you have any interest in Lorenzian manifolds as instances of pseudo-Riemannian manifolds,
then this might be of interest, especially for the theorem of Campbell:
"The embedding of General Relativity in five dimensions."
Carlos Romero, Reza Tavakol, Roustam Zalaletdinov.
General Relativity and Gravitation.
March 1996, Volume 28, Issue 3, pp 365-376. (Springer link.)

Abstract.
  We argue that General Relativistic solutions can always be locally embedded in Ricci-flat 5-dimensional spaces. This is a direct consequence of a theorem of Campbell (given here for both a timelike and spacelike extra dimension, together with a special case of this theorem) which guarantees that any $n$-dimensional Riemannian manifold can be locally embedded in an $(n+1)$-dimensional Ricci-flat Riemannian manifold. [...]

And there are many papers in some sense following, e.g.: "The embedding of space–times in five dimensions with nondegenerate Ricci tensor,"
F. Dahia and C. Romero, J. Math. Phys. 43, 3097 (2002). (AIP link.)
A: See here:
MR0262980  Reviewed Greene, Robert E. Isometric embeddings of Riemannian and pseudo-Riemannian manifolds. Memoirs of the American Mathematical Society, No. 97 American Mathematical Society, Providence, R.I. 1970 iii+63 pp. (Reviewer: W. F. Pohl)
