Dirac structures and generalized complex structures on TM × ℝh

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We consider Courant and Courant-Jacobi brackets on the stable tangent bundle TM × h of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on TM × h can be prolonged to TM × 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, 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 a are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal almost contact structure (F, Z, ). 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.

Original languageEnglish
Pages (from-to)453-474
Number of pages22
JournalAdvances in Geometry
Issue number3
StatePublished - 20 Jul 2007


  • Courant bracket
  • Dirac structure
  • Generalized complex structure
  • Normal generalized almost contact structure

ASJC Scopus subject areas

  • Geometry and Topology


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