Generalized contact structures

Abstract

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the three-dimensional Heisenberg group and its cocompact quotients.

Approaches to Generalize Contact Structures

“…Given Definition 3.3, we expect a deformation theory to allow a contact structure, as an almost cosymplectic structure, to deform to an almost contact structure. We will develop such a deformation in a forthcoming paper [18].…”

“…Proof: As a generalized almost contact structure, a classical almost contact structure is given by θ = 0 and π = 0. The map Φ in this case is given by (18).…”

Generalized Kählerian manifolds and transformation of generalized contact structures

“…For more details, we refer the reader to references [14], [22], [26]. Let M be a m-dimensional smooth manifold, the space of sections of the vector bundle T M ⊕ T * M −→ M is endowed with the following R-bilinear operations.…”

Generalized contact bundles

“…Several approaches to odd-dimensional analogues of generalized complex structures can be found in the literature [13,22,18,19,1]. They are often named generalized contact structures and all of them include contact structures globally defined by a contact 1-form.…”