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.