# Ricci curvature of embedded spacetime

1. If I am not mistaken, there is a theorem which states that every Riemannian manifold can be embedded in the $n$-dimensional Euclidean space for some large-enough $n$.

2. Does it also hold for preudo-Riemannian manifolds and Minkowski spacetimes?

3. If so, any curved spacetime of GR can be embedded in some $n$-dimensional Minkowski spacetime. The induced metric is $g_{\mu \nu} = \partial_{\mu} \phi^A \partial_{\nu} \phi_A$, where $\phi^A$ are $n$-dimensional coordinates.

4. It seems extremely hard to calculate the Ricci curvature scalar of such embedding because the inverse metric $g^{\mu \nu}$ depends non-polynomial on partial derivatives of 'fields' $\phi^A$. But still, this Ricci curvature can be evaluated.

My question is: could you provide a reference to some work where it is evaluated? Maybe it can be done in the $\det g_{\mu \nu} = 1$ gauge? After all I am interested in the Tailor expansion of the Einstein-Hilbert action with respect to derivatives of $\phi^A$ fields.