The longitudinal KAM-cocycle of a magnetic flow
  Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\Omega$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la \Omega$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form. 
Gabriel P. Paternain