On exact Poisson structures

By studying the exactness of multi-linear vectors on an orientable smooth manifold M, we give some characterizations to exact Poisson structures defined on M and study general properties of these structures. Following recent works [12, 13, 15], we will pay particular attention to the classification of some special classes of exact Poisson structures such as Jacobian and quasihomogeneous Poisson structures. A characterization of exact Poisson structures which are invariant under the flow of a class of completely integrable systems will also be given.