Closed differential one-form ($\eta \in {\Omega }^1(M),d\eta =0$)

A differential one-form is closed when its exterior derivative vanishes. In coordinates, $\eta ={\eta }_{i}d{x}^i$ is closed exactly when ${\partial }_i{\eta }_j={\partial }_j{\eta }_i$. The Poincaré lemma makes it locally an exact differential form, but global exactness can fail through nonzero integrals around closed curves.

Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the Poincaré lemma supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of de Rham cohomology.

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.