Let $M$ be a connected smooth manifold, let $x \in M$, let $G$ be a Lie group, and let
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\begin{align*}
q: \widetilde M \to M
\end{align*}
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be the universal covering map. Choose a lift $\tilde x \in q^{-1}(x)$, and identify the deck transformation group of $q$ with $\pi_1(M,x)$ by the inverse-endpoint convention: if a loop representing $\alpha \in \pi_1(M,x)$ is lifted starting at $\tilde x$, then its endpoint is $\alpha^{-1}\tilde x$.
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Then isomorphism classes of smooth principal right $G$-bundles
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\begin{align*}
\pi: P \to M
\end{align*}
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equipped with flat smooth principal connections, where isomorphisms are connection-preserving principal bundle isomorphisms over $\operatorname{id}_M$, are in natural bijection with conjugacy classes of Lie group homomorphisms
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\begin{align*}
\rho: \pi_1(M,x) \to G.
\end{align*}
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More precisely, if $p \in P_x$ is chosen, parallel transport defines a homomorphism $\rho_p: \pi_1(M,x) \to G$ by