Dirac Manifolds

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 ⊕ V satisfying a maximality condition, and having the property that a certain symmetric form on V ⊕ V 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 ⊕ TP satisfying pointwise the properties of the linear case. If a bundle L ⊂ TP ⊕ TP 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 dΩ 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.