Trivial fiber bundle

A smooth fiber bundle $\pi:E\to M$ with typical fiber $F$ is trivial (or globally trivial) if there exists a bundle isomorphism

covering $\mathrm{id}_M$, i.e. $\mathrm{pr}_1\circ \Psi=\pi$. Equivalently, $E$ admits a single global trivialization, so that all transition functions can be chosen to be the identity.

Triviality is a global property: every fiber bundle is locally a product by definition, but global triviality can fail because the local product charts may glue nontrivially.

Examples

1. Product bundles: $\mathrm{pr}_1:M\times F\to M$ is trivial by construction.
2. Euclidean tangent bundle: $T\mathbb{R}^n\to\mathbb{R}^n$ is trivial via the standard identification $T\mathbb{R}^n\cong \mathbb{R}^n\times\mathbb{R}^n$.
3. Nontrivial example: the Möbius line bundle over $S^1$ is a smooth fiber bundle with typical fiber $\mathbb{R}$ but is not trivial.