Rigidity properties of Anosov optical hypersurfaces

Abstract

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