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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Univ. Paris 11 (Orsay)
Joint work with Abdelghani Zeghib
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).
degenerated and Y is isotropic.
Remark: if [sol,sol] is non-degenerated and Y,Z are isotropic then the metric is flat.
(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.