Three-dimensional Lorentz Geometries

1. Three-dimensional Lorentz Geometries Sorin Dumitrescu Univ. Paris 11 (Orsay) Joint work with Abdelghani Zeghib

2. Klein geometries • Definition: A Klein geometry (G,X=G/H) is a simply connected space X endowed with a transitive action of a Lie group G. • If the G-action preserves some riemannian (lorentzian) metric on X the geometry is called riemannian(lorentzian).

3. Manifolds locally modelled on Klein geometries • Definition: A manifold M is locally modelled on a (G,X)-geometry if there is an atlas of M consisting of local diffeomorphisms with open sets in X and where the transition functions are given by restrictions of elements of G.

4. Examples • X= : flat riemannian geometry • X= : Minkowski space • X= • X= : anti de Sitter space (of constant negative sectional curvature)

5. Maximality • A riemannian (lorentzian) geometry (G,X=G/H) is maximal if G is of maximal dimension among the Lie groups acting transitively on X and preserving a riemannian (lorentzian) metric. • Remark: riemannian (lorentzian) geometries of constant sectional curvature are maximal.

6. Classification • Theorem: If M is a compact threefold locally modeled on a (G,X)-lorentzian (non riemannian) geometry then: • If (G,X) is maximal, then (G,X) is one of the following geometries : Minkowski, anti-de Sitter, Lorentz-Heisenberg or Lorentz-SOL. • Without hypothesis of maximality, X is isometric to a left invariant metric on one of the following groups:

7. Lorentz-Heisenberg The Lie algebra heis : [X,Y]=[X,Z]=0,[Y,Z]=X. Three classes of metrics : If the norm of X=0: the metric is flat. If the norm of X=-1: geometry of riemannian kind. If the norm of X=1: Lorentz-Heisenberg (non riemannian maximal geometry).

8. Lorentz-SOL • Lie algebra sol: • [X,Y]=Y,[X,Z]=-Z,[Y,Z]=0; • RY⊕RZ=[sol,sol] • Metric Lorentz-SOL: [sol,sol] is degenerated and Y is isotropic. Remark: if [sol,sol] is non-degenerated and Y,Z are isotropic then the metric is flat.

9. Completness • Definition: M locally modelled on a (G,X)-geometry is complete if the universal covering of M is isometric to X • Remark: then M=Γ\X, where Γ is a discrete subgroup of G.

10. Geodesic completness • Lemma: M is a locally modelled on a (G,X)-geometry and the G-action on X preserve some connexion ∇. If the connexion inherited on M is geodesically complete then the (G,X)-geometry of M is complete. Corollary: If M is compact and (G,X) is riemannian then (G,X)-geometry of M is complete.

11. Lorentz completness • Theorem: Any compact threefold locally modelled on a (G,X)-Lorentz geometry is complete. • Remark: X is not always geodesically complete.

12. Uniformization • Theorem: If M is a compact threefold endowed with a locally homogeneous Lorentz metric with non compact (local) isotropy group then M admits Lorentz metrics of (non-negative) constant sectional curvature.

13. Holomorphic Riemannian Metrics • Theorem: If a complex compact threefold M admits some holomorphic riemannian metric then it admits one with constant sectional curvature. • Remark: any holomorphic riemannian metric on M is locally homogeneous.