On PSL(2,ℂ) and on the space of geodesics of ℍ 3 as holomorphic Riemannian manifolds

— We discuss some geometric aspects of PSL(2,C), SL(2,C), and the space G of the geodesics of H3 equipped with some suitable structures of Riemannian holomorphic manifolds of constant sectional curvature. We also observe that G is a symmetric space for the group PSL(2,C) and use it to deduce some correlations between their holomorphic Riemannian metrics.