Space of spacelike embeddings as infinite-dimensional manifold

Question

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?

Answer 1

A standard reference on infinite dimensional manifolds is

Kriegl, Andreas; Michor, Peter W., The convenient setting of global analysis, Mathematical Surveys and Monographs. 53. Providence, RI: American Mathematical Society (AMS). x, 618 p. (1997). ZBL0889.58001. (on author's website)

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 $\mathcal{M}$.