Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\ast}g$ becomes a Riemannian metric on $\Sigma$.


In a paper it was, without references, stated the space


$$\mathrm{Emb}^{\infty}(\Sigma,\mathcal{M}):=\{i:\Sigma\to\mathcal{M}\mid i\text{ smooth spacelike embedding such that }h=i^{\ast}g\text{ is Riemannian} \}$$


is a manifold. Does anyone know any references where this fact is proven? Also, what type of infinite-dimensional manifold is it? I would expect that it is an open subset of $C^{\infty}(\Sigma,\mathcal{M})$ (?) and hence, it should be a Frechet manifold. Also, what about "differentiable" structures on this manifold?



Space of spacelike embeddings as infinite-dimensional manifold
A standard reference on infinite dimensional manifolds is


*Kriegl, Andreas; Michor, Peter W.*, [**The convenient setting of global analysis**](http://www.ams.org/online_bks/surv53/), Mathematical Surveys and Monographs. 53. Providence, RI: American Mathematical Society (AMS). x, 618 p. (1997). [ZBL0889.58001](https://zbmath.org/?q=an:0889.58001). [(on author's website)](https://www.mat.univie.ac.at/%7Emichor/apbookh-ams.pdf)


Some notes on the space of embeddings between two manifolds are in §44.1. Your space of spacelike embeddings $\mathrm{Emb}^\infty(\Sigma,\mathcal{M})$ is itself an open subset of the smooth embeddings between $\Sigma$ and $\mathcal{M}$.


