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Transactions of the American Mathematical Society

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
MathSciNet review: 998124
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Abstract: A Dirac structure on a vector space $ V$ is a subspace of $ V$ with a skew form on it. It is shown that these structures correspond to subspaces of $ V \oplus {V^{\ast}}$ satisfying a maximality condition, and having the property that a certain symmetric form on $ V \oplus {V^{\ast}}$ 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 $ O(V)$. 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 $ P$, which are defined to be smooth subbundles of the bundle $ TP \oplus {T^{\ast}}P$ satisfying pointwise the properties of the linear case. If a bundle $ L \subset TP \oplus {T^{\ast}}P$ defines a Dirac structure on $ P$, then we call $ L$ a Dirac bundle over $ P$. A $ 3$-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 $ [B,B]$ and $ d\Omega $ respectively. A function $ f$ on a Dirac manifold is called admissible if there is a vector field $ X$ such that the pair $ (X,df)$ is a section of the Dirac bundle $ L$; the pair $ (X,df)$ 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 $ Q$ of a Dirac manifold $ P$. The induced bracket on admissible functions on $ Q$ is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.

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Keywords: Poisson manifold, symplectic manifold, Dirac brackets, constrained dvnamics, Lie algebroid
Article copyright: © Copyright 1990 American Mathematical Society