dg.differential geometry – Soft structure in the space of sections of a group of fiber and gauge sets

Leave $$xi$$ be a fiber package $$F hookrightarrow E a B$$ (where each space is smooth, T2 and second countable), leave $$Gamma ( xi)$$ Be the space of smooth sections. We can complete $$Gamma ( xi)$$ with respect to a Sobolev $$(l, 2)$$-Norma and get the space of Sobolev sections. $$H_l ( xi)$$.

I read that $$H_l ( xi)$$ it can be given the structure of a Hilbert collector (see the examples of Uhlenbeck and Freed and four collectors), and that the space tangent to a section $$s in H_l ( xi)$$ is given by $$H_l (s ^ * mathcal {V} xi)$$ (here $${ mathcal {V}} xi$$ is the vertical beam of $$xi$$ which is a sub-bundle of $$TE$$).
It is not difficult to take a curve in $$H_l ( xi)$$ and discover who the tangent space is, but the book does not accurately describe the uniform structure in $$H_l ( xi)$$.

I would like to see how a graph of $$H_l ( xi)$$ or $$Gamma ( xi)$$ It seems. Or a more precise definition of the smooth structure in these spaces.

Motivations

1. The spaces $$C ^ infty (M, N), H_l (M, N)$$ they are particular cases when $$E = M times N$$ Y $$B = M$$.
2. The group gauge $$mathcal {G}$$ of a $$G$$-primary package $$G hookrightarrow P to M$$ is the group of automorphisms (as a main package) of $$P$$, you can identify with $$Gamma (M, P times _ { text {Ad}} G)$$. It is useful to know that the Lie algebra of $$mathcal {G}$$ identifies with $$Gamma (M, P times _ { text {ad}} mathfrak {g})$$.

Expectations

If we consider our first example 1. of $$C ^ infty (M, N)$$, $$M$$ Y $$N$$ both are metric spaces (correct a Riemannian metric for simplicity), therefore $$C ^ infty (M, N)$$ Naturally it is a metric space.
Intuitively, given a map. $$f$$, Describe all the maps of a neighborhood with the help of the exponential map and a vector field of $$N$$ along $$f$$, that is, 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 modeling a neighborhood of $$f$$ with vector fields throughout $$f$$ that are in the domain of the exponential map (if $$N$$ Compact so that the injectivity radius is positive, but what if $$N$$ It is not?). This I hope to be a variety of Frechèt.
One problem is showing that we can get all the neighborhoods of $$f$$ With this construction.

In the most general case of sections of a fiber bundle, I would consider similar vector fields in $$E$$ That they are vertical so that the previous construction conserves the fiber.

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