I am currently reading [this paper](http://projecteuclid.org/euclid.jdg/1214442469) in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most common" differentiable structure on $Gauge(P)$ would be.


Thank you very much!



Differentiable structure on the Gauge group of a principal bundle?
For *compact* base manifold and reasonably well behaved structure groups $G$ (like a finite-dimensinoal or a Banach-Lie group) there is the structure of a Frechet-Lie group on the group $Gauge(P)$. This is worked out in Chapter 1 of [Lie Group Structures on Symmetry Groups of Principal Bundles](http://dx.doi.org/10.1016/j.jfa.2007.05.016) (what I have called "property SUB" there is satisfied under the above conditions on $G$). It is not stated explicitly that $Gauge(P)$ is modelled on a Frechet space, but the modelling space is a closed subspace of a finite product of Frechet spaces (Proposition 1.4) and thus a Frechet space.  
 The same is also true for the group $Aut(P)$ of all bundle automotphisms of $P$, see Theorem 2.14. The unifying theme here is that these groups can all be seen as groups of bisections in locally trivial Lie groupoids (work in progress).  
  

If you are interested in smooth maps *into* $Gauge(P)$, then you can take the following criterion: let $f\colon I\to Gauge(P)$ be a map and let $f\_i\colon I\to C^\infty(\overline{V\_i},G)$ be the corresponding maps with respect to a chosen local trivialsation $\Phi\_i\colon P|\_{U\_i}\to U\_i\times G$ (such that $\overline{V\_i}$ is a manifold with corners and $P|\_{\overline{V\_i}}$ is also trivial to be precise). Then $f$ is smooth if and only if all $f\_i$ are smooth. This is due to the fact that $Gauge(P)$ is diffeomorphic to a submanifold of $\prod\_{i=1}^n C^\infty(\overline{V\_i},G)$ (which in turn is is implicitly contained in the proof of Theorem 1.11). For the smoothness of maps into mapping spaces there exist many useful criteria, like smoothness of pull-back, push-forward and composition maps.


