Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0402361

Coupling Poisson and Jacobi structures on foliated manifolds

Izu Vaisman
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 manifolds. Then we discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Izu Vaisman. 2004-02-23. Coupling Poisson and Jacobi structures on foliated manifolds. https://doi.org/10.1142/s0219887804000307

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗