Dirac manifolds

A Dirac structure on a vector space V V is a subspace of V V with a skew form on it. It is shown that these structures correspond to subspaces of V ⊕ V ∗ V ⊕ {V} satisfying a maximality condition, and having the property that a certain symmetric form on V ⊕ V ∗ 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 ) 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 P , which are defined to be smooth subbundles of the bundle T P ⊕ T ∗ P TP ⊕ {T}P satisfying pointwise the properties of the linear case. If a bundle L ⊂ T P ⊕ T ∗ P L ⊂ TP ⊕ {T}P defines a Dirac structure on P P , then we call L L a Dirac bundle over P P . A