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Given any Riemannian or Semi-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 $g^\prime$ ?

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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.

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For the Lorentzian case see http://arxiv.org/abs/0812.4439. In this article the authors give conditions for the existence of the imbedding.

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