Associated bundle - Knowlpedia

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

The associated bundle with fiber $F$ is the quotient

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

where the equivalence relation is

(p\cdot g,\, f)\sim (p,\, g\cdot f)\qquad (p\in P,\; f\in F,\; g\in G).

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

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

is well-defined, and $P\times_G F$ is a smooth fiber bundle over $M$ with typical fiber $F$ .

Concretely, $P\times_G F$ is a bundle of orbits: it is obtained from the product $P\times F$ by dividing out the diagonal $G$ -action determined by the right action on $P$ and the left action on $F$ . When $F$ is a vector space with a linear action, this specializes to an associated vector bundle.

Structure group

If local sections of $P$ satisfy $s_j=s_i g_{ij}$ , the induced fiber coordinates on $P\times_G F$ change by $(x,f)\mapsto(x,g_{ij}(x)\cdot f)$ . These maps give the associated bundle its G-bundle structure, with model fiber $F$ .

Examples

Tangent bundle from frames. If $P=\mathrm{Fr}(M)$ and $F=\mathbb{R}^n$ with the standard left action of $GL(n)$ , then $P\times_G F$ is canonically isomorphic to the tangent bundle $TM$ .
Line bundles from $U(1)$ -bundles. If $G=U(1)$ and $F=\mathbb{C}$ with the usual multiplication action, then $P\times_{U(1)}\mathbb{C}$ is a complex line bundle whose unit circle bundle recovers $P$ .
Adjoint bundle. Taking $F=G$ with conjugation action produces the adjoint bundle $\mathrm{Ad}(P)$ (a bundle of groups over $M$ ).