Coupling Tensors and Poisson Geometry Near a Single Symplectic Leaf

Abstract

In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on fiber bundles, called coupling tensors. We show that every Poisson structure near a closed symplectic leaf can be realized as a coupling tensor. Our main result is a geometric criterion for the neighborhood equivalence between Poisson structures over the same leaf. This criterion gives a Poisson analog of the relative Darboux theorem due to Weinstein. Within the category of the algebroids, coupling tensors are introduced on the dual of the isotropy of a transitive Lie algebroid over a symplectic base. As a basic application of these results, we show that there is a well defined notion of a ``linearized'' Poisson structure over a symplectic leaf which gives rise to a natural model for the linearization problem.