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Foliation-coupling Dirac structures. (English) Zbl 1109.53076

The author shows that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf.
The Dirac structures were first defined by Courant and Weinstein. In this paper, the author concentrates on the case where the manifold is endowed with a regular foliation and extends the notion of coupling from Poisson structures to Dirac structures.
Furthermore, the corresponding generalization of the Vorobjev characterization of coupling Poisson structures is obtained. The coupling condition is finally used along a submanifold, instead of a foliation. As a byproduct, an invariant which recalls the second fundamental form of a submanifold of a Riemannian manifold is found.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
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