Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $\Pi$. We say
that $M$ is b-Poisson if the map $\Pi^n:M\to\Lambda^{2n}(TM)$ intersects the
zero section transversally on a codimension one submanifold $Z\subset M$. This
paper will be a systematic investigation of such Poisson manifolds. In
particular, we will study in detail the structure of $(M,\Pi)$ in the
neighbourhood of $Z$ and using symplectic techniques define topological
invariants which determine the structure up to isomorphism. We also investigate
a variant of de Rham theory for these manifolds and its connection with Poisson
cohomology.Comment: 34 pages. Some changes have been implemented mainly in Sections 2 and
6. Minor changes in exposition. References have been adde
We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of the proof parallels the proof of the symplectic case, but the rest of the proof is quite different, since we are naturally led to using the calculus of polyvector fields, rather than differential forms; in particular, we use in the end a Poisson version of the classical Carathéodory-Jacobi-Lie theorem, which we also prove. At the end of the article, we generalize the action-angle theorem to the setting of non-commutative integrable systems on Poisson manifolds.
ContentsSince F m is compact, it is covered by finitely many of the sets V m ′ , say V m 1 , . . . , V m ℓ . Thus, if every pair of the diffeomorphisms φ m 1 , . . . , φ m ℓ agrees on the intersection of their domain of definition (whenever non-empty), we
Abstract. We consider an integrable Hamiltonian system with n-degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a Gequivariant way, to the linearized foliation in a neighborhood of a compact singular non-degenerate orbit. We also show that the non-degeneracy condition is not equivalent to the non-resonance condition for smooth systems.Résumé. On considère un système hamiltonien intégrableà n degrés de liberté et une action symplectique d'un groupe de Lie compact G qui laisse invariantes les intégrales premières. On prouve que le feuilletage lagrangien singulier attachéà ce système hamiltonien est symplectiquementéquivalent, de façon Gequivariante, au feuilletage linearisé dans un voisinage d'une orbite compacte singulière. On démontre aussi que la condition de non-dégénéréscence n'est paséquivalenteà la non-résonance pour les systèmes différentiables.
Dedicated to the memory of Paulette Libermann whose cosymplectic manifoldsplay a fundamental role in this paper.Abstract. In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we will see in [GMP].
Abstract. We study Hamiltonian actions on b-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.