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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
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