Threedimensional Lorentz Geometries

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Three-dimensional Lorentz Geometries

• Sorin Dumitrescu
• Univ. Paris 11 (Orsay)

Joint work with Abdelghani Zeghib
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Klein geometries

• Definition A Klein geometry (G,XG/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).

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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.

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Examples

• X flat riemannian geometry
• X Minkowski space
• X
• X anti de Sitter space (of constant negative
  sectional curvature)

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Maximality

• A riemannian (lorentzian) geometry (G,XG/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.

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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

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Lorentz-Heisenberg

• The Lie algebra heis X,YX,Z0,Y,ZX.
• Three classes of metrics
• If the norm of X0 the metric is flat.
• If the norm of X-1 geometry of riemannian kind.
• If the norm of X1 Lorentz-Heisenberg (non
  riemannian maximal geometry).

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Lorentz-SOL

• Lie algebra sol
• X,YY,X,Z-Z,Y,Z0
• RY?RZsol,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.

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Completness

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

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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.

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Lorentz completness

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

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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.

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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.