# Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, does it have a Banach manifold structure?

If $M$ is not compact, does the space of $C^k$($k<\infty$) Riemannian metrics have a Banach manifold structure?

I need some references about those problems.

• I don't know anything about noncompact manifolds, but this is the space of $C^k$ sections of a certain fiber bundle. In the compact case (just as with spaces of smooth maps) this is a very general setting that has a Banach manifold structure for $k<\infty$. I'm not sure what the standard reference for these sort of questions is. (Probably something written by Peter Michor.) – user98602 Nov 27 '15 at 4:59