Smooth fiber bundle - Knowlpedia

Let $M$ , $E$ , and $F$ be smooth manifolds, with $F$ nonempty. A smooth map $\pi:E\to M$ is a smooth fiber bundle with typical fiber $F$ if:

$\pi$ is surjective, and
for every $x\in M$ there exists an open neighborhood $U\ni x$ and a diffeomorphism $\Phi:\pi^{-1}(U)\longrightarrow U\times F$ such that $\mathrm{pr}_1\circ \Phi=\pi|_{\pi^{-1}(U)}$ .

Trivializations and consequences

Such a $\Phi$ is a local trivialization of the bundle over $U$ , and $F$ is called the typical fiber (model fiber). A family of compatible local trivializations covering $M$ forms a bundle atlas, and on overlaps the change of trivialization is encoded by a transition function.

The local product condition implies that $\pi$ is a submersion, so every smooth fiber bundle is a fibered manifold.

Examples

Trivial bundle: $M\times F\to M$ with projection $\mathrm{pr}_1$ is a smooth fiber bundle with typical fiber $F$ .
Tangent bundle: $\tau:TM\to M$ is a smooth fiber bundle with typical fiber $\mathbb{R}^{\dim M}$ , using coordinate charts on $M$ to build trivializations.
Principal bundles: every principal G-bundle $P\to M$ is a smooth fiber bundle with typical fiber $G$ (with additional group-action structure).