The moduli space of (pseudo)-Riemannian metrics $g$ on a given space (manifold) $X$.
On the site CartSp of smooth Cartesian spaces consider the sheaf
which sends each $U \in CartSp$ to the set of Riemannian metrics on $U$.
Then let $X$ be a smooth manifold, or more generally a diffeological space, or more generally a Lie groupoid or more generally a smooth ∞-groupoid, all regarded in $\mathbf{H} =$ Smooth∞Grpd.
Write
for the concretification of the internal hom: the space of metrics on $X$. (Various variations and extensions of this statement are of interest and can easily be written out. The above direct statement works for possibly degenerate metrics.)
A point in this space is a single (pseudo-)Riemannian metric on $X$.
The group $Aut(X)$ of automorphisms of $X$ acts on this by precomposition in the natural way
If $X$ is a smooth manifold then $Aut(X) = Diff(X)$ is the group of diffeomorphisms of $X$.
The quotient (action groupoid, moduli stack)
is the moduli space of (pseudo-)Riemannian metrics on $X$.
Various variations of this are of interest. For instance there one consider the Einstein-Hilbert action
The critical locus of this function is the moduli space of Einstein metrics.
For $X$ a smooth manifold, $Met(X)$ itself is a contractible space.
In the context of the theory of gravity, the moduli space of pseudo-Riemannian metrics on $X$ is the configuration space of the field theory of gravity (general relativity). The moduli space of Einstein metrics is then called the covariant phase space: this is the subspace of solutions of the Einstein equations.
Textbook references include
chapter 4 of
A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. 10, Springer-Verlag, New York, (1987) MR 88f:53087 Zbl 0613.53001
M. E. Shanks, The Space of Metrics on a Compact Metrizable Space , American Journal of Mathematics Vol. 66, No. 3, Jul. (1944) (JSTOR)
Halldor Eliasson, On variations of metrics Math. Scand. 29 (1971) 317-327 (pdf)
F. Farrell, Pedro Ontaneda, The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible (pdf)
Boris Botvinnik, Bernhard Hanke, Thomas Schick, and Mark Walsh, Homotopy groups of the space of metrics of positive scalar curvature (pdf)