Given any Riemannian or [Semi-Riemannian manifold](http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold) $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n copies of "$-1$" and m "$+1$" such that $(M,g)$ is embedded in $(E,g^\prime)$ and $g$ is the induced metric of $g^\prime$ ?



Is there some Riemannian manifold's version of Whitney theorem?
Yes, have a look at Robert Greene's book *Isometric Embeddings of Riemannian and Pseudo Riemannian Manifolds,* Volume 97 of Memoirs of the American Mathematical Society
Memoirs, 1970.


