, for any , is automatically Poisson. The Cartesian product of two Poisson manifolds and is again a Poisson manifold. Let be a (regular) foliation of dimension on and a closed foliation two-form for which the power is nowhere-vanishing. This uniquely determines a regular Poisson structure on by requiring the symplectic leaves of to be the leaves of equipped with the induced symplectic form . Let be a Lie group acting on a Poisson manifold by Poisson diffeomorphisms. If the action is free and proper, the quotient manifold inherits a Poisson structure from (namely, it is the only one such that the submersion is a Poisson map). Poisson cohomology The Poisson cohomology groups of a Poisson manifold are the cohomology groups of the cochain complex where the operator is the Schouten-Nijenhuis bracket with . Notice that such a sequence can be defined for every bivector on ; the condition is equivalent to , i.e. being Poisson. Using the morphism , one obtains a morphism from the de Rham complex to the Poisson complex , inducing a group homomorphism . In the nondegenerate case, this becomes an isomorphism, so that the Poisson cohomology of a symplectic manifold fully recovers its de Rham cohomology. Poisson cohomology is difficult to compute in general, but the low degree groups contain important geometric information on the Poisson structure: is the space of the Casimir functions, i.e. smooth functions Poisson-commuting with all others (or, equivalently, smooth functions constant on the symplectic leaves); is the space of Poisson vector fields modulo Hamiltonian vector fields; is the space of the infinitesimal deformations of the Poisson structure modulo trivial deformations; is the space of the obstructions to extend infinitesimal deformations to actual deformations. Modular class The modular class of a Poisson manifold is a class in the first Poisson cohomology group, which is the obstruction to the existence of a volume form invariant under the Hamiltonian flows. It was introduced by Koszul and Weinstein. Recall that the divergence of a vector field with respect to a given volume form is the function defined by . The modular vector field of a Poisson manifold, with respect to a volume form , is the vector field defined by the divergence of the Hamiltonian vector fields: . The modular vector field is a Poisson 1-cocycle, i.e. it satisfies . Moreover, given two volume forms and , the difference is a Hamiltonian vector