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For a codimension-one tangent distribution, a coorientation is an orientation of its normal quotient line bundle. An oriented real line bundle is trivial: local positive sections can be combined with a partition of unity without cancellation. Hence a cooriented distribution is the kernel of a global nowhere-vanishing one-form. This choice is independent of the orientation of the ambient manifold, and reversing it does not change the Godbillon-Vey invariant.

A coorientation gives a globally nonvanishing $C^2$ one-form $\zeta$ with $E=\ker \zeta$. For tangent fields $A,B$ in the integrable distribution,since integrability makes their bracket tangent to $E$. Thus $\zeta \wedge d\zeta =0$. Choose a transverse field $Y$ with $\zeta (Y)=1$. The $C^1$ one-form $\eta =-{\iota }_{Y}d\zeta$ then satisfies $d\zeta =\eta \wedge \zeta$, as is checked on $E$ and $Y$.

The Godbillon-Vey invariant in the unnormalized convention isThe three-form is continuous at the specified regularity, so its integral is defined. We prove independence of both choices by producing exact differences with $C^1$ primitives, allowing ordinary Stokes theorem.

First, for a fixed $\zeta$, another admissible auxiliary form has the form ${\eta }^{\prime }=\eta +a\zeta$. From $d^2\zeta =0$ and $d\zeta =\eta \wedge \zeta$, one gets $d\eta \wedge \zeta =0$. Expanding the two three-forms, using this relation and $\zeta \wedge d\zeta =0$, givesIts integral on the closed manifold vanishes.

Next any other defining form is ${\zeta }^{\prime }=h\zeta$ with $h$ nowhere zero. Let $g=\log |h|$. Thenso one may use ${\eta }_0^{\prime }=\eta +dg$. Its three-form differs byAgain the difference integrates to zero; any subsequent auxiliary choice is handled by the first calculation. This includes reversing the chosen coorientation, since $h$ can be negative. Thus the invariant, and the corresponding de Rham cohomology class, depends only on the oriented manifold and the foliation defined by $E$. Reversing the orientation of $N$, unlike reversing the coorientation of $E$, changes the sign of its numerical value.

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.