To every $\alpha\in T^{\ast}_{b}B$ we can associate a vector field $v_{\alpha}$ on $F_{b}$ by

$$\iota_{v_{\alpha}}\omega=f^{\ast}\alpha.$$

Let $\phi_{\alpha}^{t}$ be the flow of $v_{\alpha}$ with time $t\in\mathbb{R}$. Then we define the action $\theta_{\alpha}$ of $\alpha$ on $F_{b}$ by

$$\theta_{\alpha}(p)=\phi_{\alpha}^{1}(p),$$

where $p\in F_{b}$. One can check that such an action is well defined and transitive. Then, $\Lambda_{b}$ defined as

$$\Lambda_{b}=\{\lambda\in T^{\ast}_{b}B\ |\ \theta_{\lambda}(p)=p,\ \text{for all}\ p\in F_{b}\}$$

is a closed discrete subgroup of $T^{\ast}_{b}B$, i.e. a lattice. From the properness of $F_{b}$ it follows that $\Lambda_{b}$ is maximal (in particular homomorphic to $\mathbb{Z}^{n}$) and that $F_{b}$ is diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$. ∎