= Averaging closed one-forms on a surface unit tangent bundle
{title2=$\zeta=\pi^*\xi+a\omega+dh,\quad d\xi=aK\Omega_a$}
Average a closed one-form over fibre rotations. The average differs from the original by an exact form because each rotation is isotopic to the identity; the <Cartan formula for the Lie derivative> supplies an explicit homotopy integral. Its vertical value $a$ is constant, since closedness and rotational invariance give $da=0$. Subtracting $a\omega$ leaves a basic form $\pi^*\xi$, and closedness gives $d\xi=aK\Omega_a$. For a zero-flux magnetic field, the Liouville mean of $\zeta(X+fV)$ is zero: the directional mean of $\xi(v)$ vanishes, the vertical mean is $2\pi a\int f\Omega_a$, and the coboundary has zero mean by volume preservation.
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