Definition:Smooth Fiber Bundle

Definition
Let $E, M, F$ be smooth manifolds.

Let $\pi$ be a smooth surjection.

Let $B = \left({ E, M, \pi, F }\right)$ be a fiber bundle with a system of local trivializations $\left\{{ \left({ U_\alpha, \chi_\alpha }\right) : \alpha \in I }\right\}$ such that $\forall \alpha \in I$ :


 * $\chi_\alpha : \pi^{-1}\left({ U_\alpha }\right) \to U_\alpha \times F$

is smooth.

Then $B$ is called a smooth fiber bundle on $M$.