Let $G$ be a finite-dimensional Lie group with Lie algebra $\mathfrak{g}$, and let $\pi:P\to M$ be a smooth right principal $G$-bundle equipped with a principal connection form $\omega\in\Omega^1(P;\mathfrak{g})$. Let $\gamma:[0,1]\to M$ be a smooth path, and let $p_0\in P$ satisfy $\pi(p_0)=\gamma(0)$. Then there exists a unique smooth path $\widetilde{\gamma}:[0,1]\to P$ such that

\begin{align*} \pi\circ\widetilde{\gamma}=\gamma, \end{align*}

\begin{align*} \widetilde{\gamma}(0)=p_0, \end{align*}

and

\begin{align*} \omega_{\widetilde{\gamma}(t)}(\dot{\widetilde{\gamma}}(t))=0 \end{align*}

for every $t\in[0,1]$.