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

Let (M,g)(M,g) be a pseudo-Riemannian manifold. If MM is compact, gg is Riemannian and the tangent bundle TMTM is equipped with the Sasaki metric gsgs, parallel vector fields are the only harmonic maps from (M,g)(M,g) to (TM,gs)(TM,gs). Critical points of the energy functional E|X(M)E|X(M), restricted to maps defined by vector fields, are again parallel vector fields. On the other hand, if gg is Lorentzian, then vector fields satisfying some harmonicity properties need not be parallel. We investigate such properties for left-invariant vector fields on three-dimensional Lorentzian Lie groups, obtaining several classification results and new examples of critical points of energy functionals.