# Smooth structure on the space of sections of a fiber bundle and gauge group

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.

• 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. – Pavel Dec 13 '18 at 13:57

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 $$\phi$$.