On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented

Atiyah class of a Manin pair

A Courant algebroid $E$ with a Dirac structure $L\subset E$ is said to be a Manin pair. We first discuss $E$-Dorfman connections on predual vector bundles $B$ and develop the corresponding Cartan calculus. This is then used in relation to Courant-Dorfman cohomology to compute a cohomology class that measures the obstruction to the existence of a compatible $E$-Dorfman connection on predual vector bundles $B$ extending a given $L$-Dorfman action on $B$.Comment: Comments are welcom

Poisson brackets with prescribed Casimirs

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.Comment: 24 page

Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,\Lambda,E)$ for which 1 is an admissible function and Jacobi quotient manifolds of $M$. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.Comment: 18 page

On the geometric quantization of twisted Poisson manifolds

We study the geometric quantization process for twisted Poisson manifolds. First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for twisted Poisson manifolds and we use it in order to characterize their prequantization bundles and to establish their prequantization condition. Next, we introduce a polarization and we discuss the quantization problem. In each step, several examples are presented

On a new relation between Jacobi and homogeneous Poisson manifolds

We establish a new, very close, relationship which links Jacobi structures and homogeneous Poisson structures defined on the same manifold and study the characteristic foliations of the related structures. Several examples of this construction are also given

Structure locale de variétés de Jacobi-Nijenhuis

After a brief review on the basic notions and the principal results concerning the Jacobi manifolds, the relationship between homogeneous Poisson manifolds and conformal Jacobi manifolds, and also the compatible Jacobi manifolds, we give a generalization of some of these results needed for the contents of this paper. We introduce the notion of Jacobi-Nijenhuis structure and we study the relation between Jacobi-Nijenhuis manifolds and homogeneous Poisson-Nijenhuis manifolds. We present a local classification of homogeneous Poisson-Nijenhuis manifolds and we establish some local models of Jacobi-Nijenhuis manifolds