Gfrktyl

Let $xi$ be a fiber bundle $Fhookrightarrow Eto B$ (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 $sin H_l(xi)$ 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 $E = Mtimes N$ and $B =M$.


2. The gauge group $mathcal{G}$ of a $G$-principal bundle $Ghookrightarrow Pto M$ is the group of automorphisms (as a principal bundle) of $P$, it can be identified with $Gamma(M,Ptimes_{text{Ad}}G )$. It is useful to know that the Lie algebra of $mathcal{G}$ is identified with $Gamma(M,Ptimes_{text{ad}}mathfrak{g})$.

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 $Xin Gamma(f^*TN)$ this should induce $g=xmapsto text{exp}_{f(x)}(X_{x})in C^infty(M,N)$. 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.

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:

• Wockel: Infinite-dimensional and higher structures in differential geometry


• Kriegl & Michor: The convenient setting of global analysis


• Hamilton: The inverse function theorem of Nash and Moser

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.