  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.
