= Vertical Maslov cycle of a surface thermostat
{title2=$\Lambda_V=\{\mathbb RF(x,v)\oplus\mathbb RV(x,v)\}$}
For a <surface thermostat>, the planes containing $F$ form the <projective bundle> of $T(SM)/\mathbb RF$. 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 $\mathbb RF+\mathbb R(V+rH)$, it is $r=0$; the lifted flow satisfies $\dot r=1$ on the cycle, hence crosses it transversely.
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