Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $\pi: P \to M$ be a smooth right principal $G$-bundle over a smooth manifold $M$. Let $\mathcal A(P)$ denote the set of smooth principal connection forms on $P$, meaning smooth $\mathfrak g$-valued $1$-forms $\omega \in \Omega^1(P;\mathfrak g)$ satisfying
for every $g \in G$, where $R_g: P \to P$ is the right action and $\xi_P$ is the fundamental vertical vector field generated by $\xi$.
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Fix $\omega_0 \in \mathcal A(P)$. Then for every $\omega_1 \in \mathcal A(P)$ there exists a unique $\alpha \in \Omega^1(M;\operatorname{ad}(P))$, identified with its corresponding horizontal and $G$-equivariant smooth form $\widetilde{\alpha} \in \Omega^1(P;\mathfrak g)$, such that
Conversely, for every $\alpha \in \Omega^1(M;\operatorname{ad}(P))$, the form $\omega_0 + \widetilde{\alpha}$ is a smooth principal connection form on $P$. Hence $\mathcal A(P)$ is an affine space modelled on the [vector space](/page/Vector%20Space) $\Omega^1(M;\operatorname{ad}(P))$.