The magnetic flow on the manifold of oriented geodesics of a three dimensional space form

Let M be the three dimensional complete simply connected manifold of constant sectional curvature 0,1 or -1. Let \mathcal{L} be the manifold of all (unparametrized) complete oriented geodesics of M, endowed with its canonical pseudo-Riemannian metric of signature (2,2) and Kähler structure J. A smooth curve in \mathcal{L} determines a ruled surface in M. We characterize the ruled surfaces of M associated with the magnetic geodesics of \mathcal{L}, that is, those curves \sigma in \mathcal{L} satisfying \nabla_{\dot{\sigma}}\dot{\sigma}=J\dot{\sigma}. More precisely: a time-like (space-like) magnetic geodesic determines the ruled surface in M given by the binormal vector field along a helix with positive (negative) torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity. This provides a relationship between the geometries of \mathcal{L} and M.