TITLE: The space of Riemannian metrics on a given manifold.
QUESTION [27 upvotes]: For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set 
$$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$
I'd like to know about the typical differentiable structures one can place on $\mathcal{Met}(M)$, and how are they constructed. More specifically, is this in general an infinite-dimensional Banach or Hilbert manifold? Or perhaps a Fréchet manifold? What if $M$ is compact?
Also, any references containing a good deal of details, proofs, etc. would be deeply appreciated. Thanks!

REPLY [4 votes]: Hamilton's 1982 paper The inverse function theorem of Nash and Moser available here and here is also a very good reference.  
The space of Riemannian metrics on a compact manifold is a Frechet manifold.