Resumen de Lorentzian Flat Lie Groups Admitting a Timelike Left-Invariant Killing Vector Field

Hicham Lebzioui

• We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flat Lie group (G, µ) admits a timelike left-invariant Killing vector field if and only if G admits a left-invariant Riemannian metric which has the same Levi-Civita connection of µ. Finally, we give an useful characterization of left-invariant pseudo-Riemannian flat metrics on Lie groups G satisfying the property: for any couple of left invariant vector fields X and Y their Lie bracket [X, Y ] is a linear combination of X and Y .