Géométries Lorentziennes de dimension 3 : classification et complétude

Abstract

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the metric has constant sectionnal curvature, or is a left invariant metric on the Heisenberg group or the Sol group. These geometries, on each of the latter two groups are characterized by having a non-compact isotropy without being flat. Recall, for the need of his formulation of the geometrization conjecture, W. Thurston counted the 8 maximal Riemannian geometries in dimension 3. Here, we count only 4 maximal Lorentz geometries, but ignoring those which are at the same time Riemannian. Also, all such manifolds are geodesically complete, except the previous non flat left invariant Lorentz metric on the SOL group.