Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

TL;DR

This paper establishes necessary and sufficient topological conditions for a three-dimensional dynamical system to admit a globally defined bi-Hamiltonian structure, linking it to the vanishing of specific characteristic classes.

Contribution

It provides a topological characterization of when a nonvanishing vector field on an orientable 3-manifold admits a global bi-Hamiltonian structure, relating it to Chern and Bott classes.

Findings

Bi-Hamiltonian structures exist iff the first Chern class of the normal bundle vanishes.

Global compatibility requires the Bott class of the associated foliation to vanish.

The results connect dynamical systems with topological invariants of manifolds.

Abstract

In this work, we show that an autonomous 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. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.