The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation

Details The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation

2002-11-26

arxiv.org/abs/math/0211405v1

Mathematics

Differential Geometry

11 pages

Scientific paper

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unimodular. Finally, we prove that the first leafwise cohomology is an invariant of Morita equivalence.

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