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

Abstract

Let ( M , g ) be a pseudo-Riemannian manifold. If M is compact, g is Riemannian and the tangent bundle T M is equipped with the Sasaki metric g s , parallel vector fields are the only harmonic maps from ( M , g ) to ( T M , g s ) . Critical points of the energy functional E | X ( M ) , restricted to maps defined by vector fields, are again parallel vector fields. 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 on three-dimensional Lorentzian Lie groups, obtaining several classification results and new examples of critical points of energy functionals.