## On the local structure of Dirac manifolds.(English)Zbl 1142.53063

Summary: We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point $$m$$ of a Dirac manifold $$M$$, there is a well-defined transverse Poisson structure to the pre-symplectic leaf $$P$$ through $$m$$. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 58H05 Pseudogroups and differentiable groupoids

### Keywords:

normal forms; Poisson structure; Dirac structure; foliations
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