2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/211285In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/Differential Geometry53C22, 53C50, 57M50text