Three-dimensional Lorentz Geometries

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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.

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### 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.
• If the G-action preserves some riemannian (lorentzian) metric on X the geometry is called riemannian(lorentzian).
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.
Examples
• X= : flat riemannian geometry
• X= : Minkowski space
• X=
• X= : anti de Sitter space (of constant negative sectional curvature)
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.
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:
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).

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.

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.
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.

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