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 be the manifold of all (unparametrized) complete oriented geodesics of M, endowed with its canonical pseudo-Riemannian metric of signature (2, 2) and Kahler structure J. A smooth curve in determines a ruled surface in M. We characterize the ruled surfaces of M associated with the magnetic geodesics of C, that is, those curves a in,C satisfying del(sigma)sigma = J 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 L and M.