On Bott-Morse Foliations and their Poisson structures in dimension three

Abstract

We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.This section follows notations used in [20] and [21], where Bott-Morse foliations on dimension 3 were described.Let M m be a closed, orientable, smooth manifold of dimension m, for m ≥ 3. Let F be a codimension-one smooth foliation with singularities on M . Denote by Sing(F) the set of singular points of F.

Compatible poisson structures on fibered 5-manifolds

“…Note that various results on low dimensional Poisson manifolds in dimensions 2, 3 and 4 were obtained, for example, in [11,9,10,4,16,8,7]. The results, presented in this paper, can be used for the classification of Poisson structures around 2-dimensional Poisson submanifolds.…”

“…Remark 4.5 If M Π ∩ supp β = Ø, then the rank of Π equals 0 or 2 and the Poisson structure Π is of the Flaschka-Ratiu type [6]. Such class appears in the problem of the construction of Poisson structures with prescribed characteristic foliations [6,8,7]. Now, we observe that, according to decomposition (4.14), equations (4.10), (4.11) for (γ, κ, β) can be represented as follows.…”

On computational Poisson geometry I: Symbolic foundations

“…The bivector field Π is called the λ-gauge transformation of Π [34,10,9]. A pair of bivector fields Π and Π on M are said to be gauge equivalent if they are related by (16) for some differential 2-form λ on M satisfying (15). If Π is a Poisson bivector field, then Π is a Poisson bivector field if and only if λ is closed along the symplectic leaves of Π.…”

“…The function gauge transformation computes the gauge transformation of a bivector field, and the determinant of the morphism in (15). Such a determinant defines the set of points of M where the gauge transformation is well defined.…”

“…The function flaschka ratiu bivector computes the Flaschka-Ratiu bivector field, with respect to Ω 0 , and the corresponding symplectic form of a 'maximal' set of scalar functions [12,18,35,15]. differential Ki ← an array encoding the differential dK i of K i…”