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 ?
Best Answer
Especially if your interested in dynamical systems, I highly recommend Abraham--Marsden--Ratiu, Manifolds, tensor analysis, and applications.
For a more Riemannian-geometric/global-analytic focus, you might want to try Klingenberg, Riemannian Geometry, or Lang, Differentiable Manifolds.
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 $N$.