Associated bundle from a principal bundle and a left G-space

Construction of the fiber bundle P×_G F associated to a principal G-bundle and a left G-space.

Associated bundle from a principal bundle and a left G-space

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

Construction (associated bundle). Consider $P\times F$ with the right $G$ -action

(p,f)\cdot g := (pg, g^{-1}\cdot f).

Define the associated bundle as the quotient

P\times_G F := (P\times F)/G,

and write $[p,f]$ for the equivalence class of $(p,f)$ . The projection

\pi_F: P\times_G F \to M,\qquad \pi_F([p,f])=\pi(p),

is well-defined and makes $P\times_G F$ into a smooth fiber bundle with typical fiber $F$ .

If $s:U\to P$ is a local section, then

\Phi_s:U\times F\to \pi_F^{-1}(U),\qquad (x,f)\mapsto [s(x),f],

is a bundle trivialization; on overlaps, the transition functions are given by the $G$ -action on $F$ .

Examples

Taking $F=G$ with left multiplication, $P\times_G G$ is canonically isomorphic to $P$ as a principal $G$ -bundle (via $[p,g]\mapsto pg$ ).
Taking $F=G$ with conjugation recovers the adjoint bundle $P\times_G G$ , a bundle of groups over $M$ .
If $F$ is a homogeneous space $G/H$ , then $P\times_G (G/H)$ is a fiber bundle with fiber $G/H$ ; reductions of structure group to $H$ can be expressed as sections of this associated bundle.