Trivial principal bundle

The product principal bundle M times G with its standard projection and right action.

Trivial principal bundle

Let $M$ be a smooth manifold and let $G$ be a Lie group . The trivial principal $G$ -bundle over $M$ is

\pi: M\times G \longrightarrow M,\qquad \pi(x,g)=x,

equipped with the right action

(x,g)\cdot h := (x,gh),\qquad h\in G.

The bundle $M\times G\to M$ is a principal G-bundle : the action is free and transitive on each fiber, and local trivializations are global (the identity map).

It has a canonical global section

s:M\to M\times G,\qquad s(x)=(x,e),

where $e\in G$ is the identity.

A principal bundle $P\to M$ is called trivial if it is isomorphic (as a principal $G$ -bundle over $M$ ) to $M\times G$ .

Examples

Any principal bundle over a contractible base is (often) trivial in practice.
For many Lie groups and typical geometric bases, contractibility of $M$ forces every principal bundle to be trivial; in particular, every principal bundle over $\mathbb R^n$ is trivial.
Gauge transformations are just maps to the group.
Every bundle automorphism of $M\times G$ covering $\mathrm{id}_M$ is of the form
$\Phi_f(x,g)=(x,f(x)g)$
for a smooth map $f:M\to G$ (a gauge transformation).
Frame bundles on parallelizable manifolds.
If $M$ admits a global frame of $TM$ , then the frame bundle $\mathrm{Fr}(TM)$ is trivial: choosing a global frame identifies it with $M\times \mathrm{GL}(n,\mathbb R)$ .