Theorem: A trivial principal bundle admits a global section

Let $\pi:P\to M$ be a principal G-bundle with structure Lie group $G$ .

If $P$ is trivial, i.e. there exists a principal bundle isomorphism

\Psi:P\xrightarrow{\cong} M\times G,

then $P$ admits a smooth global section. Concretely, if $e\in G$ is the identity, then

s(x):=\Psi^{-1}(x,e)

defines a smooth section $s:M\to P$ with $\pi\circ s=\mathrm{id}_M$ .

Equivalent characterizations

Equivalently, triviality of $P$ is characterized by the existence of a global section, together with the converse implication.

Examples

Canonical section of the product. For $P=M\times G$ , the section $x\mapsto (x,e)$ is smooth and globally defined.

Trivializations differ by gauge transformations. If $\Psi$ and $\Psi'$ are two trivializations, the associated sections differ by right multiplication by a smooth map $M\to G$ .

Pullback of a trivial bundle. If $f:N\to M$ is a smooth map and $P\cong M\times G$ , then the pullback bundle $f^*P$ is trivial and inherits a global section by pulling back $x\mapsto(x,e)$ .