Extension of structure group

Let $P\to M$ be a principal G-bundle with structure group $G$ , and let $\varphi:G\to H$ be a smooth homomorphism of Lie groups.

The extension of structure group of $P$ along $\varphi$ is the quotient

P\times_\varphi H := (P\times H)/\sim,

where

(p\cdot g,\; h)\sim (p,\; \varphi(g)\,h).

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

[p,h]\longmapsto \pi(p),

and the right $H$ -action is

[p,h]\cdot k := [p,hk].

With these structures, $P\times_\varphi H\to M$ is a principal $H$ -bundle.

This construction is a special case of an associated bundle: it is the associated bundle to $P$ with fiber $H$ where $G$ acts on $H$ by left multiplication through $\varphi$ .

There is a canonical smooth bundle map

P\to P\times_\varphi H,\qquad p\mapsto [p,e],

covering $\mathrm{id}_M$ . It is $\varphi$ -equivariant: $[pg,e]=[p,\varphi(g)]=[p,e]\cdot\varphi(g)$ . For $\varphi=\operatorname{id}_G$ , this is a principal $G$ -bundle morphism in the fixed-group sense.

Examples

From oriented orthonormal frames to oriented frames. The inclusion $SO(n)\hookrightarrow GL^+(n)$ extends the principal $SO(n)$ -bundle of oriented orthonormal frames to the principal $GL^+(n)$ -bundle of oriented frames. (Using $O(n)\hookrightarrow GL(n)$ gives the full frame bundle.)
Nonzero vectors in a line bundle. Extending a principal $U(1)$ -bundle along $U(1)\hookrightarrow \mathbb{C}^\ast$ produces the principal $\mathbb{C}^\ast$ -bundle of nonzero vectors in the associated complex line bundle.
From SU(n) to U(n). Extending a principal $SU(n)$ -bundle along $SU(n)\hookrightarrow U(n)$ yields a principal $U(n)$ -bundle; geometrically this forgets the determinant-one constraint.