Bundle of orbits

Let $\pi:P\to M$ be a principal bundle with right principal action of the Lie group $G$, and let $F$ be a smooth left $G$-space.

The bundle of orbits associated to $(P,F)$ is the orbit space

for the right $G$-action on $P\times F$ defined by

When this action is free and proper (as it is in the principal-bundle setting with smooth $F$), the orbit space is a smooth manifold and the natural projection

makes it a smooth fiber bundle over $M$. This orbit bundle is canonically identified with the associated bundle $P\times_G F$ defined via the equivalent relation $(p\cdot g,f)\sim(p,g\cdot f)$.

Examples

1. Fiber a point. If $F=\{\ast\}$ with the trivial action, then $(P\times F)/G \cong P/G \cong M$.
2. Trivial principal bundle. If $P=M\times G$, then $(P\times F)/G \cong M\times F$ by sending the orbit of $(x,h,f)$ to $(x,h\cdot f)$.
3. Recovering P. If $F=G$ with left multiplication, then $(P\times G)/G \cong P$ via $[(p,h)]\mapsto p\cdot h$.