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I was wondering if there are any existence results similar to the Nash embedding theorem where a manifold with a Riemannian metric is isometrically embedded into flat Minkowski space, with metric of signature $(-,+,\ldots,+)$. Allowing for the target space to be of Lorentzian signature certainly seems to help with negatively curved Riemannian metrics, i.e. the two dimensional hyperbolic plane has no isometric embedding in $\mathbb{R}^3$, however, it is easily embedded into Minkowski space $\mathbb{R}^{2,1}$ as a surface of constant proper time away from the origin. The question would be given a Riemannian manifold of dimension $n$, what is the minimal dimension of Minkowski space $\mathbb{R}^{d-1,1}$ in which the surface is isometrically embeddable?

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  • $\begingroup$ Possibly of interest: Can every Riemannian manifold be embedded in a sphere? $\endgroup$ – Andrew D. Hwang Aug 1 '17 at 0:46
  • $\begingroup$ Your last question is unknown even in the case of isometric embeddings/immersions into Euclidean spaces. What do you expect as an answer? An "explicit" integer-valued function on the set of isometry classes of all Riemannian manifolds? $\endgroup$ – Moishe Kohan Aug 5 '17 at 1:54
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$\newcommand{\Reals}{\mathbf{R}}$The answer is "yes" because, as you note, hyperbolic $d$-space embeds isometrically in $\Reals^{d,1}$ as the set of future-pointing unit timelike vectors, and a horosphere in hyperbolic space is a Euclidean $(d - 1)$-space. Offhand I don't know whether this (together with the Nash embedding theorem) is optimal.

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