# Nash Embedding Theorems for Pseudo-Riemannian Manifolds?

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

• 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 '16 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 '16 at 12:37

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. [...]