Magnetic X-ray transform

The magnetic transform integrates functions or velocity-evaluated one-forms along magnetic trajectories, rather than ordinary geodesics. For closed trajectories, every exact one-form integrates to zero. Hyperbolic dynamics can make these the entire degree-one kernel.

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.