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On the local structure of Dirac manifolds

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  01 May 2008

Jean-Paul Dufour  and
Aïssa Wade
Jean-Paul Dufour
Affiliation:
Département de Mathématiques, CNRS-UMR 5030, Université Montpellier 2, 34095 Montpellier cedex 05, France (email: dufourj@math.univ-montp2.fr)
Aïssa Wade
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (email: wade@math.psu.edu)
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Abstract

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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.

Keywords

normal formsPoisson structureDirac structurefoliations

MSC classification

Secondary: 53D17: Poisson manifolds; Poisson groupoids and algebroids
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 3 , May 2008 , pp. 774 - 786
Copyright
Copyright © Foundation Compositio Mathematica 2008