Contact isotropic realisations of Jacobi manifolds

Abstract

This paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are contact realisations of minimal dimension. These arise in the study of integrable Hamiltonian systems on contact manifolds, while also extending the Boothby-Wang construction of regular contact manifolds. The main results of the paper are local smooth and contact normal forms for contact isotropic realisations, which, amongst other things, provide an intrinsic proof of the existence of local action-angle coordinates for integrable Hamiltonian systems, as well as a cohomological criterion to construct such realisations. Moreover, one of the smooth invariants of such realisations is interpreted as providing a type of transversal projective structure on the foliation of the underlying Jacobi structures.