Bi-Hamiltonian structure of a unit geodesic vector field on a 3D space of constant negative curvature.

In this work we consider the Riemannian manifold defined by the product of an integral curve of a Cauchy-Riemann vector field on the Poincaré upper half-plane and its image in the tangent bundle. We show that for a Cauchy-Riemann vector field the Chern-Simons three-form identically vanishes and for the Killing vector field X = x ∂ x + y ∂ y the manifold is a space of constant negative curvature. We also show that the components of the connection 1-form θ define compatible Poisson structures iff θ ∧ d θ is s o (3) -valued. By virtue of this we obtain a bi-Hamiltonian structure of a unit geodesic vector field on the manifold. [ABSTRACT FROM AUTHOR]