Generalized Contact Structures Research Articles

Weakly regular Jacobi structures and generalized contact bundles

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure.

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

Conformal Changes of Odd-Dimensional Generalized Structures

In this paper, we consider the integrability of generalized almost contact and contact manifolds after conformal changes. We also study conditions under which the generalized almost contact and normal generalized contact structures, be normal after conformal changes.

Generalized Kählerian manifolds and transformation of generalized contact structures

The aim of this paper is two-fold. First, new generalized Kahler manifolds are constructed starting from both classical almost contact metric and almost Kahlerian manifolds. Second, the transformation construction on classical Riemannian manifolds is extended to the generalized geometry setting.

Generalized contact bundles

In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [8], it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

On the characterization of infinitesimal symmetries of the relativistic phase space

The phase space of relativistic particle mechanics is defined as the first jet space of motions regarded as time-like one-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally a generalized contact structure on the odd-dimensional phase space. In the paper, infinitesimal symmetries of the phase structures are characterized. More precisely, it is proved that all phase infinitesimal symmetries are special Hamiltonian lifts of distinguished conserved quantities on the phase space. It is proved that generators of infinitesimal symmetries constitute a Lie algebra with respect to a special bracket. A momentum map for groups of symmetries of the geometric structures is provided.

Generalized contact structures

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 co-compact quotients.

Lie bialgebroids of generalized CRF-manifolds

The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008) [6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form ξ ± η , where ξ is a vector field and η is a 1-form on M with η ( ξ ) = 1 . It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form ξ ± η , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a Cauchy–Riemann structure on a manifold.

Prolongation of Tensor Fields and G-Structures in Tangent Bundles of Second Order

Tangent and cotangent bundles have been defined and studied by Yano, Ishihara, Patterson and others. Duggal gives the notion of GF-structure, which plays an important role in the differentiable manifold [1]. R. Nivas and Ali have studied the existence of GF-structure and generalized contact structure on the tangent bundle and some interesting results have been obtained for such structures [2]. Prolongation of tensor fields, almost complex and almost product structures have been defined and studied by Yano, Ishihara [3] and others whereas Das [4] and Morimoto [5] have studied the prolongation of F-structure and G-structures respectively to the tangent bundles. In the present paper problems of prolongation in tangent bundle of second order and few results on GF, fa(3, −1) and generalized contact structures have been discussed.

The geometry of the bundle of connections

A generalized symplectic structure on the bundle of connections $p\colon C(P)\rightarrow M$ of an arbitrary principal G-bundle $\pi\colon P\rightarrow M$ is defined by means of a $p^{\ast}\mathrm{ad}P$ -valued differential 2-form $\Omega_{2}$ on C(P), which is related to the generalized contact structure on $J^{1}(P)$ . The Hamiltonian properties of $\Omega_{2}$ are also analyzed.

Complex contact structures and the first eigenvalue of the dirac operator on Kähler manifolds

In this paper Kählerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kähler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kählerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kählerian Killing spinors are obtained.