On the $C^k$-embedding of Lorentzian manifolds in Ricci-flat spaces

In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area, and then motivate both the mathematical and physical interest in this problem. We show that any $n$-dimensional compact Lorentzian manifold $(M^{n},g)$, with $g$ in the Sobolev space $H_{s+3}$, $s>\frac{n}{2}$, admits an isometric embedding in an $(2n+2)$-dimensional Ricci-flat semi-Riemannian manifold. The sharpest result available for this type of embeddings, in the general setting, comes as a corollary of Greene's remarkable embedding theorems [R. Greene, Mem. Am. Math. Soc. 97, 1 (1970)], which guarantee the embedding of a compact $n$-dimensional semi-Riemannian manifold into an $n(n+5)$-dimensional semi-Euclidean space, thereby guaranteeing the embedding into a Ricci-flat space with the same dimension. The theorem presented here improves this corollary in $n^{2}+3n-2$ codimensions by replacing the Riemann-flat condition with the Ricci-flat one from the beginning. Finally, we will present a corollary of this theorem, which shows that a compact strip in an $n$-dimensional globally hyperbolic space-time can be embedded in a $(2n+2)$-dimensional Ricci-flat semi-Riemannian manifold.