Typical fiber

Let $\pi:E\to M$ be a smooth fiber bundle . A typical fiber (or model fiber) is a smooth manifold $F$ for which there exists an open cover $\{U_i\}$ of $M$ and local trivializations $\Phi_i:\pi^{-1}(U_i)\to U_i\times F$.

In particular, for every $x\in M$ the fiber $E_x=\pi^{-1}(x)$ is diffeomorphic to $F$, via restriction of $\Phi_i$ to $\{x\}\times F$. The typical fiber is not part of the bare fibered-manifold structure; it is extra data specifying which manifold is used as the local model. If both $F$ and $F'$ can serve as typical fibers for the same bundle, then $F$ and $F'$ must be diffeomorphic.

Examples

1. Tangent and cotangent bundles: for an $n$-manifold $M$, the typical fiber of $TM\to M$ (and of $T^*M\to M$) is $\mathbb{R}^n$.
2. Principal bundles: the typical fiber of a principal $G$-bundle $P\to M$ is the Lie group $G$ itself.
3. Sphere bundles: the unit sphere bundle $S(E)\to M$ of a rank-$k$ vector bundle has typical fiber $S^{k-1}$.