Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space other than OH2 , and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)K¨ahler structure on L(M).
Alekseevsky, D.V., Guilfoyle, B. & Klingenberg, W. On the geometry of spaces of oriented geodesics. Ann Glob Anal Geom 40, 389–409 (2011). https://doi.org/10.1007/s10455-011-9261-5