Godbillon-Vey invariant ($\mathrm{gv}(E)={\int }_N\eta \wedge d\eta$)

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 $-d(ad\zeta )$. 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.