Some geometrical properties of the Oscillator group

We consider the oscillator group equipped with a bi-invariant Lorentzian metric. Some geometrical properties of this space and the harmonicity properties of left-invariant vector fields on this space are determined. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Left-invariant vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.