Some geometrical properties of the Oscillator group

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also determined, and the energy of critical points is explicitly calculated.