Principal H-subbundle

Let $P\to M$ be a principal G-bundle with structure group a Lie group $G$, and let $H\subset G$ be a Lie subgroup.

A principal H-subbundle of $P$ is a submanifold $Q\subset P$ such that:

1. $\pi(Q)=M$ and $\pi|_Q:Q\to M$ is a surjective submersion,
2. $Q$ is preserved by the restricted right action of $H$, i.e. $q\cdot h\in Q$ for all $q\in Q$, $h\in H$,
3. the $H$-action on $Q$ is free and transitive on the fibers of $\pi|_Q$.

Then $(Q,\pi|_Q,M,H)$ is a principal $H$-bundle, and the inclusion $Q\hookrightarrow P$ is $H$-equivariant. Giving such a subbundle is precisely the concrete form of a reduction of structure group from $G$ to $H$.

Examples

1. Orthonormal frames. For a Riemannian manifold, the orthonormal frame bundle is a principal $O(n)$-subbundle of the full frame bundle (a $GL(n)$-bundle).
2. Trivial bundle subgroups. If $P=M\times G$, then $Q:=M\times H$ is a principal $H$-subbundle.
3. Unit circle subbundle of a line bundle. For a Hermitian complex line bundle $L\to M$, the unit circle bundle $S(L)$ is a principal $U(1)$-subbundle of the principal $\mathbb{C}^\ast$-bundle of nonzero vectors in $L$.