Dirac Structures and Generalized Complex Structures on TM × R h by Izu Vaisman

We consider Courant and Courant-Jacobi brackets on the stable tangent bundle TM ×R of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on TM × R can be prolonged to TM × R, 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 = 1, ..., h), where P is a bivector field, θ is a 2-form, F is a (1, 1)-tensor field, Za are vector fields and ξ 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 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. 1 Brackets on stable tangent bundles All the manifolds and mappings of the present note are assumed of the C∞ class. In differential topology, the vector bundle TM × R, where M is a 2000 Mathematics Subject Classification: 53C15, 53D17.