Let $G$ be a Lie group, let $H \le G$ be a closed Lie subgroup, let $M$ be a smooth manifold, and let $\pi: P \to M$ be a smooth principal right $G$-bundle. Let $\bar{\pi}: P/H \to M$ denote the smooth quotient bundle obtained from the restricted right action of $H$ on $P$, equivalently the associated bundle $P \times_G (G/H) \to M$ for the left action of $G$ on $G/H$.

Then reductions of $P$ to $H$, meaning smooth principal right $H$-subbundles $Q \subset P$ over $M$ whose inclusion is $H$-equivariant and satisfies $Q_m \subset P_m$ for every $m \in M$, are in bijection with smooth sections $\sigma: M \to P/H$ of $\bar{\pi}$. The bijection sends a reduction $Q \subset P$ to the section $\sigma_Q$ defined by

\begin{align*} \sigma_Q(m) = [u]_H \quad \text{for any } u \in Q_m, \end{align*}

where $[u]_H$ denotes the right $H$-orbit of $u$ in $P$. Its inverse sends a section $\sigma: M \to P/H$ to the subset

\begin{align*} Q_\sigma = \{u \in P : [u]_H = \sigma(\pi(u))\}. \end{align*}