Proposition 3.2 (Arnold-Liouville) . [04HZ]

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Proposition 3.2 (Arnold-Liouville).

In the above situation, for every $b\in B$ , $T^{\ast}_{b}B$ acts transitively on $F_{b}$ . In particular there exists a maximal sub-lattice $\Lambda_{b}$ of $T^{\ast}_{b}B$ such that $F_{b}$ is naturally diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$ , therefore $F_{b}$ is an $n$ -torus.