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 $0,1$ 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 $M$.