G\'eom\'etrie des groupes oscillateurs et vari\'et\'es localement   sym\'etriques
  The Oscillator Groups,$\G_\lambda,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant affine structure and prove that it is not Lorentzian. As an application, we provide new examples of compact pseudo-Riemannian (sometimes Lorentzian) locally symmetric, occasionally affine, manifolds, complete or incomplete. 
Shirley Bromberg and Alberto Medina