Smooth fiber bundle

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

1. $\pi$ is a surjective submersion (so $(E,\pi,M)$ is a fibered manifold ), and
2. 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 $\mathrm{pr}_1\circ \Phi=\pi|_{\pi^{-1}(U)}$.

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 .

Examples

1. Trivial bundle: $M\times F\to M$ with projection $\mathrm{pr}_1$ is a smooth fiber bundle with typical fiber $F$.
2. 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.
3. Principal bundles: every principal G-bundle $P\to M$ is a smooth fiber bundle with typical fiber $G$ (with additional group-action structure).