Global Compatibility of Bi-Hamiltonian Structures on Three Dimensional Manifolds

2023/05/29 by Beste Madran, Madran, Beste, Ender Abadoğlu +1
Mathematics · Physics and Astronomy · #53D17 #53D35 #Advanced Differential Geometry Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2305.18595

openalex publication_date 2023/05/29 · openalex created_date 2023/06/01 · openalex updated_date 2026/07/28

Abstract

It is shown in \citeyazar6 that a dynamical system defined by a nonvanishing vector field on an orientable three dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes, and the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the nonvanishing vector field vanishes. In this work, we constructed a dynamical system on S³, which is globally bi-Hamiltonian, but the Hamiltonians are not globally compatible.

Related

• Bi-Hamiltonian structure as a shadow of non-Noether symmetry
• A Simple Way of Making a Hamiltonian System Into a Bi-Hamiltonian One
• Existence of Hamiltonian Structure in 3D
• On novel Hamiltonian descriptions of some three-dimensional non-conservative systems
• On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type