CONICET | Buscador de Institutos y Recursos Humanos

Let M be the three dimensional complete simply connected manifold of constant sectional curvature 0, 1 or −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) and Kahler structure J. A smooth curve in L determines a ruled surface in M. We characterize the ruled surfaces of M associated with the magnetic geodesics of L, that is, those curves σ in L satisfying ∇σ σ = J σ. 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 .