Dirac Structures and Generalized Complex Structures on TM×\mathdsR^h

2006/07/08 by Izu Vaisman, Vaisman, Izu
Mathematics · #Advanced Operator Algebra Research #Algebraic and Geometric Analysis #Spectral Theory in Mathematical Physics #math.DG #math.SG #msc:53C15 #msc:53D17

paper · pdf · doi:10.48550/arxiv.math/0607216

LaTex, 27 pages

arxiv created 2006/07/08 · arxiv updated 2009/12/01

Abstract

We consider Courant and Courant-Jacobi brackets on the stable tangent bundle TM×\mathdsR^h of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on TM×\mathdsR^h can be prolonged to TM×\mathdsR^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,Z_a,ξ^a) (a=1,...,h), where P is a bivector field, θ is a 2-form, F is a (1,1)-tensor field, Z_a 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 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.

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