Nontrivial principal bundle with no global section

Let $\pi:P\to M$ be a principal G-bundle .

Triviality criterion via sections

A smooth global section $s:M\to P$ (so $\pi\circ s=\mathrm{id}_M$) trivializes $P$:

• Define $\Phi:M\times G\to P,\qquad \Phi(x,g):=s(x)\cdot g.$
• Then $\Phi$ is a $G$-equivariant diffeomorphism covering $\mathrm{id}_M$.
• Hence $P$ is isomorphic to the trivial principal bundle $M\times G$.

Conversely, the trivial bundle $M\times G\to M$ always has the canonical section $x\mapsto(x,e)$.

So:

Counterexample (no global section)

The Hopf bundle $\pi:S^3\to S^2$ is a principal $U(1)$-bundle that admits no global smooth section, and therefore is nontrivial. Concretely, a section would give a $U(1)$-equivariant identification $S^3\cong S^2\times U(1)$, contradicting the known topology of $S^3$ and the nontrivial clutching of the bundle.

Examples

1. Hopf fibration.
   The Hopf fibration has no global section, so it is not isomorphic to $S^2\times U(1)$.

2. Circle base with disconnected structure group.
   Using the clutching construction over the circle with $G=\mathrm{O}(1)=\{\pm1\}$ and gluing element $-1$ produces a nontrivial principal bundle over $S^1$; it has no global section.

3. Trivial bundles always have sections.
   For any $M$ and $G$, the section $x\mapsto(x,e)$ exhibits $M\times G\to M$ as trivial.