Extensions maximales et classification des tores lorentziens munis d'un champ de Killing

We define a family of model spaces for 2-dimensional Lorentzian geometry, consisting of simply connected inextendable Lorentzian surfaces admitting a Killing field. These spaces, called " universal extensions " , are constructed by an extension process and characterized by symmetry and completeness conditions. In general, these surfaces have a rich combinatorics and admit many quotient spaces and many divisible open sets. As applications, we show the existence of plenty (both topologically and geometrically) of Lorentzian surfaces with a Killing field. We also prove uniformisation results for the compact case and for the analytic case, which in particular allows us to give a classification of Lorentzian tori and Klein bottles with a Killing field