Left invariant semi Riemannian metrics on quadratic Lie groups (Q6654897)

This paper examines the flatness of left invariant semi-Riemannian metrics on quadratic Lie groups -- those which admit a bi-invariant semi-Riemannian metric. Some of these metrics can be constructed from solutions of the Yang-Baxter equation [\textit{M. Boucetta} and \textit{A. Medina}, J. Geom. Phys. 61, No. 12, 2309--2320 (2011; Zbl 1226.53070)]. Regardless, known examples of such metrics are scarce. While the case of flat left invariant Riemannian metrics has been previously investigated, the non-definiteness of a semi-Riemannian metric makes the present investigation more challenging [\textit{J. W. Milnor}, Adv. Math. 21, 293--329 (1976; Zbl 0341.53030)].\N\NThis paper identifies a geometric condition on Lie groups that is necessary and sufficient to guarantee the existence of a flat left invariant semi-Riemannian metric on quadratic Lie groups: such a metric must be geodesically complete. Using this condition, among others, the authors present numerous examples of such metrics and demonstrate cases when such a metric does not exist.