# A book on Banach Manifold for a Dynamicist

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 ?

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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 $N$.

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The way Lang makes a differentiable structure on $C^0(M,N)$ is via the exponential map, in the case of Riemann manifolds, and via sprays for more general manifolds. Is this the standard way you mentioned? – Pietro Majer Nov 13 '11 at 22:31
@Pietro: Yes, that's what I meant. – Brian Clarke Nov 14 '11 at 5:18

The first textbook I thought is: Palais, The Foundations of Global Non-linear Analysis, Benjamin-Cummings, 1968.

Finally you could look here at Graff's review of The Metric theory of Banach Manifolds'' by Ethan Atkin, for many other references.

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