The space of Riemannian metrics on a given manifold.

Solution 1:

As far as I understand the situation, the $C^\infty$-topology makes the space of metrics a Fréchet space but if $M$ is not compact the set $\mathcal{Met}(M)$ is not open in it, and a problem of finding a convenient topology arises.

I found that a very good survey of the question is given in the first part of D.E.Blair's "Spaces of Metrics and Curvature Functionals" (Chapter 2 of "Handbook of Differential Geometry", Vol.1, Ed. by F.J.E.Dillen and L.C.A.Verstraelen, Elsevier, 2000).

As for the paper cited by @AlexE it would be convenient to have P.W. Michor's "Manifolds of differentiable mappings" at hand when reading on this subject. It is available on the author's site. One can find there all the necessary foundations including the construction of topologies.

I would love to know more on this subject too.

Solution 2:

You may want to have a look at the paper

Gil-Medrano, Michor: The Riemannian Manifold of all Riemannian Metrics, Quarterly Journal of Mathematics (Oxford) 42 (1991), 183-202

and the references therein.

Also available online at http://www.mat.univie.ac.at/~michor/rie-met.pdf.

Solution 3:

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.