The longitudinal KAM-cocycle of a magnetic flow

Abstract

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 $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form.

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