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Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with respect to a Sobolev $(l,2)$-norm and obtain the space of Sobolev sections $H_l(\xi)$.

I have read that $H_l(\xi)$ can be given the structure of an Hilbert manifold (see Uhlenbeck and Freed's Instantons and four manifolds), and that the tangent space to a section $s\in H_l(\xi)$ is given by $H_l(s^*\mathcal{V}\xi)$ (here ${\mathcal{V}}\xi$ is the vertical bundle of $\xi$ wich is a subbundle of $TE$). It is not difficult to take a curve in $H_l(\xi)$ and find out who is the tangent space but the book doesn't describe precisely the smooth structure on $H_l(\xi)$.

I would like to see how a chart of $H_l(\xi)$ or $\Gamma(\xi)$ looks like. Or a more precise definition of the smooth structure on these spaces.

Motivations

  1. The spaces $C^\infty(M,N), H_l(M,N) $ are particular cases when $E = M\times N$ and $B =M$.
  2. The gauge group $\mathcal{G}$ of a $G$-principal bundle $G\hookrightarrow P\to M$ is the group of automorphisms (as a principal bundle) of $P$, it can be identified with $\Gamma(M,P\times_{\text{Ad}}G )$. It is useful to know that the Lie algebra of $\mathcal{G}$ is identified with $\Gamma(M,P\times_{\text{ad}}\mathfrak{g})$.

Expectations

If we consider our first example 1. of $C^\infty(M,N)$, $M$ and $N$ are both metric spaces (fix a Riemannian metric for simplicity), thus $C^\infty(M,N)$ naturally is a metric space. Intuitively, given a map $f$, I would describe all the maps in a neighborhood with the help of the exponential map and a vector field of $N$ along $f$, i.e. given $X\in \Gamma(f^*TN)$ this should induce $g=x\mapsto \text{exp}_{f(x)}(X_{x})\in C^\infty(M,N)$. So we would end up modelling a neighborhood of $f$ with vector fields along $f$ that lie in the domain of the exponential map (if $N$ compact so that the injectivity radius is positive but what if $N$ is not?). This I expect to be a Frechèt manifold. One problem is to show that we can obtain all the neighborhood of $f$ with this construction.

In the more general case of sections of a fiber bundle, I would consider similarly vector fields on $E$ that are vertical so that the above construction preserve the fiber.

I expect that with the same argument but changing the topology on $C^\infty(M,N)$ we would end up with different local models, i.e. if we consider the $C^k$ metric it will be locally Banach, if we choose the $C^\infty$ metric it will be Frechèt with the Heine-Borel property and if we choose a Sobolev norm it will be not complete.

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    $\begingroup$ There is the lecture notes (Chapter 2). I believe that the construction can be adjusted for fiber bundles by taking a neighborhood of a section which looks like a vector bundle and a metric on it such that geodesics in the vertical direction remain in the fiber. $\endgroup$ – Pavel Dec 13 '18 at 13:57
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Your intuition is right. To endow the space of sections of a fiber bundle $F$ with a manifold structure at $\phi \in \Gamma^\infty(F)$ you consider a tubular neighborhood (respecting the fiber structure) about the image of $\phi$ in $F$. The tube diffeomorphism serves as a linearization of every section sufficiently close to $\phi$.

This construction is described, for example, in:

Since you mention the group of gauge transformations, the article "The Lie group of automorphisms of a principle bundle" by Abbati, Cirelli, Manià & Michor might be of interest to you.

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