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

Abstract

Let M be the three dimensional complete simply connected manifold of constant sectional curvature 0,10,1 or −1−1. Let L be the manifold of all (unparametrized) complete oriented geodesics of M, endowed with its canonical pseudo-Riemannian metric of signature (2,2)(2,2) and Kähler structure J. A smooth curve in L determines a ruled surface in M. We characterize the ruled surfaces of MM associated with the magnetic geodesics of LL, that is, those curves σσ in LL satisfying ∇σ˙σ˙=Jσ˙∇σ˙σ˙=Jσ˙. 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.