Gauge group action on connections by pullback

Let $G$ be a Lie group and let $\pi:P\to M$ be a principal G-bundle . Write $\mathrm{Conn}(P)$ for the set of principal connections on $P$, and let $\mathrm{Gauge}(P)$ denote the gauge group, i.e. the group of $G$-equivariant diffeomorphisms $\Phi:P\to P$ covering the identity on $M$ (so $\pi\circ\Phi=\pi$ and $\Phi(p\cdot g)=\Phi(p)\cdot g$).

Proposition (pullback action on $\mathrm{Conn}(P)$)

For every $\Phi\in \mathrm{Gauge}(P)$ and every principal connection $\omega\in \mathrm{Conn}(P)$, the pullback

is again a principal connection on $P$. Moreover, this defines a left group action of $\mathrm{Gauge}(P)$ on $\mathrm{Conn}(P)$, i.e.

Equivalently, if $\omega$ is viewed as a $G$-equivariant horizontal distribution $H=\ker\omega\subset TP$, then $\Phi$ sends horizontals to horizontals:

so the gauge group acts on the set of horizontal distributions defining connections.

Examples

1. Trivial bundle $P=M\times G$. A gauge transformation is given by a smooth map $g:M\to G$ via $\Phi_g(x,h)=(x,g(x)h)$. Writing a connection in a global trivialization as a $\mathfrak g$-valued 1-form $A\in\Omega^1(M;\mathfrak g)$, the action becomes $A \longmapsto A^g := \mathrm{Ad}_{g^{-1}}A + g^{-1}dg.$
2. Abelian structure group. If $G$ is abelian (e.g. $U(1)$), then $\mathrm{Ad}_{g^{-1}}$ is trivial and the transformation law reduces to $A \longmapsto A + g^{-1}dg,$ i.e. gauge transformations act by translation by an exact 1-form (in a trivialization).
3. Frame bundle viewpoint. If $P$ is a frame bundle of a vector bundle, a gauge transformation is a change of frame covering $\mathrm{id}_M$. Pulling back the connection corresponds to the usual transformation rule for connection matrices under a change of frame.