Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface $\mathrm{\Sigma }$ in the cotangent bundle $\tau :T^{\ast }M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $\mathrm{\Sigma }$ is Anosov, then a smooth 1-form $\theta$ on $M$ is exact if and only ${\tau }^{\ast }\theta$ has zero integral over every closed characteristic of $\mathrm{\Sigma }$. This result is derived from a related theorem about magnetic flows which generalizes our work in \cite{DP}. Other rigidity issues are also discussed.