Bayrakdar, T.2024-06-122024-06-1220240393-04401879-1662https://doi.org/10.1016/j.geomphys.2024.105115https://hdl.handle.net/20.500.14551/19615In this work we consider the Riemannian manifold defined by the product of an integral curve of a Cauchy-Riemann vector field on the Poincare 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 theta x + y theta y the manifold is a space of constant negative curvature. We also show that the components of the connection 1 -form theta define compatible Poisson structures iff theta perpendicular to d theta is so(3)-valued. By virtue of this we obtain a bi-Hamiltonian structure of a unit geodesic vector field on the manifold. (c) 2024 Elsevier B.V. All rights reserved.en10.1016/j.geomphys.2024.105115info:eu-repo/semantics/closedAccessCauchy-Riemann Vector FieldSpace Of Constant Negative CurvatureBi-Hamiltonian StructureUnit Geodesic Vector FieldFormsArticle198N/AWOS:0011754531000012-s2.0-85183492359Q2