1 / 13

Three-dimensional Lorentz Geometries

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

Download Presentation

Three-dimensional Lorentz Geometries

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related