 Averaging closed one-forms on a surface unit tangent bundle

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 ${\pi }^{\ast }\xi$, 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.