$M$ 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.