Let $\pi: P \to M$ be a smooth principal $G$-bundle over a smooth manifold $M$, where $G$ is a Lie group with Lie algebra $\mathfrak g$. Let $\omega \in \Omega^1(P;\mathfrak g)$ be a connection $1$-form, and let $\Omega \in \Omega^2(P;\mathfrak g)$ be its curvature form. Let $U_i,U_j \subset M$ be open sets, and let
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\begin{align*}
s_i: U_i \to P
\end{align*}
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and
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\begin{align*}
s_j: U_j \to P
\end{align*}
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be smooth local sections of $\pi$. Suppose the transition function
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\begin{align*}
g_{ij}: U_i \cap U_j \to G
\end{align*}