Coupling Poisson and Jacobi structures on foliated manifolds

Abstract

Let M be a differentiable manifold endowed with a foliation F. A Poisson structure P on M is F-coupling if the image of the annihilator of TF by the sharp-morphism defined by P is a normal bundle of the foliation F. This notion extends Sternberg's coupling symplectic form of a particle in a Yang-Mills field. In the present paper we extend Vorobiev's theory of coupling Poisson structures from fiber bundles to foliations and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. Then we discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

LateX, 38 pages

References in corpus (2)

• Some linear Jacobi structures on vector bundles
• Hamiltonian structures on foliations

Cited by in corpus (9)

• Poisson structures on almost complex Lie algebroids
• Nonlinearizability of certain Poisson structures near a symplectic leaf
• Invariant Poisson Realizations and the Averaging of Dirac Structures
• Jacobi Structures in ${\mathbb{R}}^3$
• Unimodularity Criteria for Poisson Structures on Foliated Manifolds
• Compatible Poisson Structures on Fibered $5$-Manifolds
• $G$-Invariant Deformations of Almost-Coupling Poisson Structures
• Bigraded cochain complexes and Poisson cohomology
• Generalized Hannay-Berry Connections on Foliated Manifolds and Applications