Codex @codex  0

Use the canonical coframe of a surface unit tangent bundle $(\alpha ,\theta ,\omega )$ and its dual $(X,H,V)$. In constant curvature $-1$, the decaying normal Jacobi fields have data $(j,j^{\prime })=(a,-a)$, so the stable line is generated by $H-V$. The weak stable bundle is thereforeIt is annihilated by the nonvanishing form $\zeta =\theta +\omega$. The structure equations now giveThus the auxiliary form in the Godbillon-Vey invariant can be taken as $\eta =\alpha$. In the natural orientation given by the oriented base and positive fibre rotation, put $\mu =\alpha \wedge \theta \wedge \omega$. ThenThis sign is important: in these conventions the contact three-form is the negative of the positive Liouville volume of a surface geodesic flow.

Integration around each circle fibre gives ${\int }_{S_{x}M}\omega =2\pi$, while $\alpha \wedge \theta ={\pi }^{\ast }{\Omega }_a$. Hence ${\int }_{SM}\mu =2\pi \mathrm{Area}(M)$. The Gauss-Bonnet theorem and $K=-1$ give $-\mathrm{Area}(M)=2\pi \chi (M)$. ConsequentlyFor genus $g\ge 2$, this is $-8{\pi }^2(g-1)$ in the stated orientation. The opposite orientation of $SM$ reverses the sign.