## Abstract

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.

Original language | English |
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Pages (from-to) | 917-938 |

Number of pages | 22 |

Journal | Journal of Geometry and Physics |

Volume | 56 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2006 |

## Keywords

- Coupling
- Dirac structure
- Normalized submanifolds
- Presymplectic leaf

## ASJC Scopus subject areas

- Mathematical Physics
- General Physics and Astronomy
- Geometry and Topology