On the local structure of Dirac manifolds

Publisher: Cambridge University Press

E-ISSN: 1570-5846|144|3|774-786

ISSN: 0010-437x

Source: Compositio Mathematica, Vol.144, Iss.3, 2008-05, pp. : 774-786

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Abstract

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.