Foliation-coupling Dirac structures

We extend the notion of “coupling with a foliation” from Poisson to Dirac structures and get the corresponding generalization of the Vorobjev characterization of coupling Poisson structures [Yu.M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie algebroids and related topics in differential geometry, Banach Center Publ., Polish Acad. Sci. (Warsaw) 54 (2001) 249-274; I. Vaisman, Coupling Poisson and Jacobi structures, Int. J. Geom. Meth. Mod. Phys. 1 (5) (2004) 607-637]. 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 [J.-P. Dufour, A. Wade, On the local structure of Dirac manifolds. arXiv:math.SG/0405257] on the transversal Poisson structure, and compute the Vorobjev 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.