I would like to know if its possible that given a principal fiber bundle $(P, \pi, M, G)$, the gauge group $GA(P)$ of all gauge transformations $f:P\to P$, and a single connection $1$-form $\omega_0 \in \Omega^1(P,\mathfrak{g})$ if I am able to obtain the whole space of connection $1$-forms by considering all $1$-forms of the type $f^*\omega_0$ for $f \in GA(P)$. I've been reading Bleecker and Hamilton for a research project I'm doing, and maybe I've missed it but I'm not entirely certain this claim was ever proven, disproven, or mentioned in these texts. If $\mathcal{C}$ is the space of connection one forms then we can consider the mapping $\mathcal{C} \times GA(P) \to \mathcal{C}$ given by $(\omega, f) \to f^*\omega$ to be a right action. I suppose my question is equivalent to asking whether or not this action is ever transitive, and if so under what circumstances. Either way, since I'm working on a variational problem I may not need to know the full space of connections but simply how to update locally.
I'm willing to make some hefty assumptions here:
- I'm fine if $M$ is connected and simply connected,
- $P$ can be a trivial bundle