# Nash Embedding Theorems for Pseudo-Riemannian Manifolds?

Are there analogs of the Nash Embedding Theorems for Pseudo-Riemannian Manifolds?

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Yes, and the proof much easier, one can explicitly can write a formula for the embedding into $\mathbb{R}^{N,N}$ – Anton Petrunin Apr 17 '13 at 0:12
@AntonPetrunin: Why not give a reference? Others might stumble upon this question through a web search and find your reference useful. – Alex M. Feb 14 at 17:28
@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. – Anton Petrunin Feb 15 at 12:37

## 2 Answers

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)

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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.)

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