= Kernel of the magnetic X-ray transform in degrees zero and one
{title2=$Fu=h(x)+\xi_x(v)\Longrightarrow h=0,\ \xi\text{ exact}$}
For a smooth Anosov magnetic flow on a closed oriented surface, a smooth function plus a smooth one-form can be a smooth flow coboundary only when the function is zero and the one-form is exact. Equivalently their sum has zero integral along every closed orbit precisely in that case, using the smooth <Livsic theorem>. This is the nontrivial magnetic transform kernel theorem. Exact one-forms give the converse directly, since $F(w\circ\pi)=dw(v)$. The theorem does not require pointwise negative Gaussian curvature.
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