The magnetic flow on the manifold of oriented geodesics of a three dimensional space form (Q379119)

``Let \(M\) be the 3-dimensional complete simply connected manifold of constant sectional curvature \(k=0\), \(1\), 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 \(M\).'' For example, for \(k=0\), \(-1\), a generic geodesic of \(\mathcal{L}\) describes a helicoid in \(M\) (see [\textit{B. Guilfoyle} and \textit{W. Klingenberg}, J. Lond. Math. Soc., II. Ser. 72, No. 2, 497--509 (2005; Zbl 1084.53017); \textit{N. Georgiou} and \textit{B. Guilfoyle}, Rocky Mt. J. Math. 40, No. 4, 1183--1219 (2010; Zbl 1202.53045)], respectively).NEWLINENEWLINENEWLINEIn this paper, the authors ``characterize the ruled surfaces of \(M\) associated with the magnetic geodesics of \(\mathcal{L}\), that is 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. These results establish a relationship between the geometries of \(\mathcal{L}\) and \(M\).''