Rational Lagrangian submanifold

For a Lagrangian submanifold $L$ in an exact symplectic manifold with primitive $\lambda$, its period subgroup is $\{{\int }_c\lambda :c\in H_1(L;\mathbb{Z})\}\subset \mathbb{R}$. It is rational when this subgroup is discrete. If it is nonzero, it is $\gamma \mathbb{Z}$ for a unique $\gamma >0$. Some conventions reserve the word rational for this nonzero case. The zero subgroup corresponds to an exact differential form $\lambda {|}_L$; its least positive period is interpreted as $+\infty$ if that endpoint convention is admitted.

For a nonzero discrete Liouville period group, this is its positive generator. In Euclidean symplectic space, the same group is obtained from signed symplectic areas of disks with boundary on $L$, by the Stokes theorem and contractibility of the ambient space. A product of circles of common radius $r$ is a rational Lagrangian submanifold with $\gamma (L)=\pi r^2$. Translation changes a primitive only by an exact form and hence leaves the period group unchanged.