Are there analogs of the Nash Embedding Theorems for PseudoRiemannian Manifolds?

1$\begingroup$ Yes, and the proof much easier, one can explicitly can write a formula for the embedding into $\mathbb{R}^{N,N}$ $\endgroup$ – Anton Petrunin Apr 17 '13 at 0:12

$\begingroup$ @AntonPetrunin: Why not give a reference? Others might stumble upon this question through a web search and find your reference useful. $\endgroup$ – Alex M. Feb 14 '16 at 17:28

$\begingroup$ @AlexM. There is an explicit construction is called Nash twist. For the given metric $g$ it produce one parameter family of embeddings $f_t$ in $\mathbb{R}^N$ with induced metric $g+t{\cdot}h$ for some fixed metric $h$. The embedding $x\mapsto (2{\cdot}f_t(x), f_{2\cdot t}(x))$ in $\mathbb{R}^{N,N}$ is isometric. I learned it from Gromov's book on partial differential relations. $\endgroup$ – Anton Petrunin Feb 15 '16 at 12:37
See here:
MR0262980 Reviewed Greene, Robert E. Isometric embeddings of Riemannian and pseudoRiemannian manifolds. Memoirs of the American Mathematical Society, No. 97 American Mathematical Society, Providence, R.I. 1970 iii+63 pp. (Reviewer: W. F. Pohl)
Not clear where you are headed with your concise question, but if you have any interest in Lorenzian manifolds as instances of pseudoRiemannian 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 365376. (Springer link.)
Abstract. We argue that General Relativistic solutions can always be locally embedded in Ricciflat 5dimensional 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 Ricciflat 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.)

$\begingroup$ I see the "locally embedded" part, but surely an $S^{3}XS^{1}$ spacetime topology needs an embedding into $\mathbb{R}^{4,2}$ to yield a global vector basis (in order to describe the parallelization thereof) $\endgroup$ – R. Rankin Feb 6 at 1:56