ISSN 1088-6850(online) ISSN 0002-9947(print)

Dirac manifolds

Author: Theodore James Courant
Journal: Trans. Amer. Math. Soc. 319 (1990), 631-661
MSC: Primary 58F05; Secondary 53C57
DOI: https://doi.org/10.1090/S0002-9947-1990-0998124-1
MathSciNet review: 998124
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Abstract: A Dirac structure on a vector space is a subspace of with a skew form on it. It is shown that these structures correspond to subspaces of satisfying a maximality condition, and having the property that a certain symmetric form on vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of . Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace.

Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold , which are defined to be smooth subbundles of the bundle satisfying pointwise the properties of the linear case. If a bundle defines a Dirac structure on , then we call a Dirac bundle over . A -tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are and respectively. A function on a Dirac manifold is called admissible if there is a vector field such that the pair is a section of the Dirac bundle ; the pair is called an admissible section. The set of admissible functions is shown to be a Poisson algebra.

A process is given for passing Dirac structures to a submanifold of a Dirac manifold . The induced bracket on admissible functions on is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.

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