2 4 Ja n 20 05 Foliation - coupling Dirac structures by Izu Vaisman

We extend the notion of " coupling with a foliation " from Poisson to Dirac structures and get the corresponding generalization of the Vorobiev characterization of coupling Poisson structures [20, 18]. We show that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf, give new proofs of the results of Dufour and Wade [9] on the transversal Poisson structure, and compute the Vorobiev structure of the total space of a normal bundle of the leaf. Finally, we use the coupling condition along a submanifold, instead of a foliation, in order to discuss submanifolds of a Dirac manifold which have differentiable, induced Dirac structures. In particular, we get an invariant that reminds the second fundamental form of a submanifold of a Riemannian manifold.