Characteristic classes of foliated surface bundles with area-preserving holonomy

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism from the symplectomophism group of the fiber to its first real cohomology. This crossed homomorphism extends the flux homomorphism defined on the identity component of the symplectomorphism group.

辛叶面填充的球余切丛

Symplectic foliated fillings of sphere cotangent bundles

研究特定叶面 manifold 的辛叶面填充，证明三球体 Reeb 叶面的球余切丛的叶面盘余切丛在吹胀和辛变形等价下唯一。

On foliated characteristic classes of transversally symplectic foliations

Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles for which these characteristic classes are non-trivial. This shows that the foliated cohomology classes of Kotschick and Morita carry information that is not merely topological.

Coupling Poisson and Jacobi structures on foliated manifolds

Let M be a differentiable manifold endowed with a foliation F. A Poisson structure P on M is F-coupling if the image of the annihilator of TF by the sharp-morphism defined by P is a normal bundle of the foliation F. This notion extends Sternberg's coupling symplectic form of a particle in a Yang-Mills field. In the present paper we extend Vorobiev's theory of coupling Poisson structures from fiber bundles to foliations and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifold…

Dirac Surfaces and Threefolds

We describe Dirac structures on surfaces and 3-manifolds. Every Dirac structure on a surface $M$ is described either by a regular 1-foliation or by a section of a circle bundle obtained as a fiberwise compactification of the line bundle $\wedge^2TM$. Every Dirac structure on a 3-manifold $M$ is either the union of a presymplectic manifold and a foliated Poisson manifold, or the union of a Poisson manifold and a foliated presymplectic manifold.

Connections in Poisson Geometry I: Holonomy and Invariants

We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by Izu Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.