Harmonicity and Minimality of vector fields on four-dimensional Lorentzian lie groups

We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields defining harmonic maps, and calculate explicitly the energy of these vector fields. Then we study the minimality of critical points for the energy functional.