Maslov cycle

Relative to a fixed Lagrangian subspace $L_0$, the Maslov cycle consists of Lagrangian subspaces $L$ with $L\cap L_0\ne 0$. In a two-dimensional symplectic space all lines are Lagrangian, and the cycle is the single line $L_0$ in the projective line. Fibrewise versions yield sections of a projective-line bundle and record passage through a prescribed vertical direction.

For a surface thermostat, the planes containing $F$ form the projective bundle of $T(SM)/\mathbb{R}F$. The vertical Maslov cycle is the section selecting the image of the vertical vector field of a surface unit tangent bundle. It is a smooth hypersurface. In the local projective chart $\mathbb{R}F+\mathbb{R}(V+rH)$, it is $r=0$; the lifted flow satisfies $\dot{r}=1$ on the cycle, hence crosses it transversely.