= Godbillon-Vey invariant
{c}
{title2=$\mathrm{gv}(E)=\int_N\eta\wedge d\eta$}
For a cooriented integrable codimension-one distribution, choose a nonvanishing defining form $\zeta$ and an auxiliary form with $d\zeta=\eta\wedge\zeta$. Changing the auxiliary form to $\eta+a\zeta$ changes $\eta\wedge d\eta$ by $-d(a\,d\zeta)$. Replacing $\zeta$ by $h\zeta$ changes the auxiliary form by $d\log|h|$, changing the three-form by the exact differential $d(-d\log|h|\wedge\eta)$. Stokes' theorem proves invariance on a closed oriented three-manifold. For the weak stable foliation of a curvature-minus-one surface geodesic flow, take $\zeta=\theta+\omega$, $\eta=\alpha$ to obtain $\mathrm{gv}=4\pi^2\chi(M)$ in the positive base/fibre orientation.
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