Convention: associated bundles use a left action on the fiber

Associated bundles combine the right principal action on a principal bundle with a left action on the model fiber.

Convention

Let $\pi:P\to M$ be a principal $G$-bundle equipped with the right action $P\times G\to P$, $(p,g)\mapsto p\cdot g$ (see right action convention ). Let $F$ be a smooth left $G$-space, i.e. a manifold with a smooth action $G\times F\to F$, $(g,f)\mapsto g\cdot f$ (compare smooth group action ).

We define the associated bundle

to be the quotient of $P\times F$ by the equivalence relation

We denote the equivalence class of $(p,f)$ by $[p,f]$. The projection

is well-defined and makes $P\times_G F\to M$ into a fiber bundle associated to P . This is the convention used in the associated bundle construction .

A useful bookkeeping consequence is the following: if local sections $s_i:U_i\to P$ define transition functions by $s_j=s_i\,g_{ij}$ (as in transition functions from local sections ), then the induced local trivializations satisfy

so the same $g_{ij}$ acts on the fiber by the given left action.

Examples

1. Associated vector bundle from a representation.
   If $\rho:G\to GL(V)$ is a representation , then $V$ is a left $G$-space via $g\cdot v:=\rho(g)v$, and $P\times_G V$ is an associated vector bundle .

2. Recovering a vector bundle from its frame bundle.
   For a rank $n$ vector bundle $E\to M$ with frame bundle $P=\operatorname{Fr}(E)$, take $F=\mathbb R^n$ with the standard left $GL(n)$-action. Then

   $$\operatorname{Fr}(E)\times_{GL(n)}\mathbb R^n \cong E$$

   canonically (this is the basic example behind frame-bundle descriptions of vector-bundle data ).

3. Hopf bundle and the Hopf line bundle.
   For the Hopf fibration $S^3\to S^2$ (a principal $U(1)$-bundle) and $F=\mathbb C$ with the usual left $U(1)$-action by scalar multiplication, the associated bundle $S^3\times_{U(1)}\mathbb C\to S^2$ is the standard nontrivial complex line bundle over $S^2$.