Some geometrical properties of the oscillator group

Document Type : Research Articles


Department of Mathematics, Payame noor University, P.O. Box 19395-3697, Tehran, Iran.


‎We consider the oscillator group equipped with‎ ‎a biinvariant 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.