Twisted isotropic realisations of twisted Poisson structures

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol’d theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in \cite{bursztyn_crainic_weinstein_zhu}. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising the main results of \cite{daz_delz}.

💡 Research Summary

The paper investigates the global geometry underlying non‑commutative integrable Hamiltonian systems defined on almost‑symplectic manifolds, with the aim of extending the well‑known Liouville‑Arnold theory to the setting of twisted Poisson (and more generally twisted Dirac) structures. The starting point is the recent observation by Balseiro, García‑Naranjo that many non‑holonomic integrable models can be described as twisted Poisson manifolds. Building on the definition of non‑commutative integrable systems on almost‑symplectic manifolds introduced by Fasso and Sansonetto, the authors recall that locally such systems admit generalized action‑angle coordinates: a neighbourhood of a regular invariant torus can be identified with a product of a torus and an open set in ℝⁿ, equipped with a 2‑form ω that is not closed but whose exterior derivative reproduces the twisting 3‑form ϕ of the underlying Poisson structure.

The central object of the paper is a twisted isotropic realisation: a smooth surjective submersion φ : (M, ω) → (P, π, ϕ) from an almost‑symplectic manifold (M, ω) to a twisted Poisson manifold (P, π, ϕ) satisfying three conditions: (i) φ is a submersion, (ii) the pull‑back of the Poisson bivector π coincides with the inverse of ω on the horizontal distribution (i.e. φπ♯ = ω♭⁻¹ ∘ dφ), and (iii) the fibres of φ are ω‑isotropic (the restriction of ω to each fibre vanishes). Moreover, the compatibility dω = φϕ guarantees that the failure of ω to be closed is exactly measured by the twisting 3‑form on the base.

To classify such realisations up to smooth isomorphism, the authors introduce two cohomological invariants. The first is a transversal class τ ∈ H¹(P;ℝ), which records the monodromy of the fibration (essentially the holonomy of the flat connection defined by the isotropic fibres). The second is a 2‑cocycle ω̃ ∈ H²(P;ℝ) representing the global part of the almost‑symplectic form. The key compatibility condition is the cohomological obstruction \