G'eom'etries Lorentziennes de dimension 3 : classification et compl'etude

摘要

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compactlocally homogeneous Lorentz 3-manifolds with non-compact (local) isotropygroup. One result is that, up to a finite cover, all such manifolds admitLorentz metrics of (non-positive) constant sectionnal curvature. If thegeometry is maximal, then the metric has constant sectionnal curvature, or is aleft invariant metric on the Heisenberg group or the Sol group. Thesegeometries, on each of the latter two groups are characterized by having anon-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, wecount only 4 maximal Lorentz geometries, but ignoring those which are at thesame time Riemannian. Also, all such manifolds are geodesically complete, except the previous nonflat left invariant Lorentz metric on the SOL group.