Dirac structures and generalized complex structures on

Abstract

Abstract We consider Courant and Courant–Jacobi brackets on the stable tangent bundle TM × ℝ h of a differentiable manifold and corresponding Dirac, Dirac–Jacobi and generalized complex structures. We prove that Dirac and Dirac–Jacobi structures on TM × ℝ h can be prolonged to TM × ℝ k , k > h, by means of commuting infinitesimal automorphisms. Some of the stable generalized complex structures are a natural generalization of the normal almost contact structures; they are expressible by a system of tensors (P, θ, F, Za , ξ a ) (a = 1, …, h), where P is a Poisson bivector field, θ is a 2-form, F is a (1, 1)-tensor field, Za are vector fields and ξ a are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal almost contact structure (F, Z, ξ). We prove that such a generalized structure projects to a generalized complex structure of a space of leaves and we characterize the structure by means of the projected structure and of a normal bundle of the foliation. Like in the Boothby–Wang theorem about contact manifolds, principal torus bundles with a connection over a generalized complex manifold provide examples of this kind of generalized normal almost contact structures.