Theorem: Global section implies a principal bundle is trivial

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

Theorem

If there exists a smooth section $s:M\to P$ (so $\pi\circ s=\mathrm{id}_M$), then $P$ is trivial: the map

is a $G$-equivariant diffeomorphism covering $\mathrm{id}_M$. Hence $P\cong M\times G$ as principal $G$-bundles.

Examples

1. Product bundle. For $P=M\times G$, the map $s(x)=(x,e)$ is a global section, and $\Phi$ is the identity.

2. Nontrivial bundles fail to have sections. The Hopf fibration $S^3\to S^2$ (a principal $U(1)$-bundle) has no global section; otherwise it would be diffeomorphic to $S^2\times U(1)$.

3. Triviality over contractible bases. If $M$ is contractible and a principal bundle admits a section (e.g. produced by additional structure), this theorem upgrades that section to an explicit trivialization by the formula $\Phi(x,g)=s(x)\cdot g$.