Hi all, 


Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian Geometry ?


To help the indication the problems I am facing in my research are for example, what are the usual conditions required to a Banach manifold to be metrizable. Is there a standard way to construct a metric on $C^r(M,M)$, where $M$ is a two-dimensional manifold ?



A book on Banach Manifold for a Dynamicist 
Especially if your interested in dynamical systems, I highly recommend [Abraham--Marsden--Ratiu, Manifolds, tensor analysis, and applications](http://www.ams.org/mathscinet-getitem?mr=960687).


For a more Riemannian-geometric/global-analytic focus, you might want to try [Klingenberg, Riemannian Geometry](http://www.ams.org/mathscinet-getitem?mr=1330918), or [Lang, Differentiable Manifolds](http://www.ams.org/mathscinet-getitem?mr=431240).


There is a standard way to construct a canonical *topology* on $C^r(M,N)$ for $M$ compact, one that turns $C^r(M, N)$ into a Banach manifold. But I don't think there is a canonical metric on $C^r(M, N)$ unless you put some additional structure on $N$. 


