Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups

Let (M, g) be a pseudo-Riemannian manifold. If M is compact, g is Riemannian and thetangent bundle TM is equipped with the Sasaki metric gs, parallel vector fields are the only harmonic maps from (M, g) to (TM, gs). On the other hand, if g is Lorentzian, then vector fields satisfying some harmonicity properties need not be parallel. We investigate such properties for left-invariant vector fields onthree-dimensional Lorentzian Lie groups, obtaining several classification results and new examples of critical points of energy functionals.