Left invariant semi Riemannian metrics on quadratic Lie groups

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left invariant metrics on non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie group admits a flat left invariant semi Riemannian metric. The case of quadratic 2 step nilpotent Lie groups is also addressed.