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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Dirac manifolds
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by Theodore James Courant PDF
Trans. Amer. Math. Soc. 319 (1990), 631-661 Request permission


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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 319 (1990), 631-661
  • MSC: Primary 58F05; Secondary 53C57
  • DOI:
  • MathSciNet review: 998124