Biharmonic vector fields on pseudo-Riemannian manifolds

The bienergy of a vector field on a pseudo-Riemannian manifold (M,g) is defined to be the bienergy of the corresponding map (M,g)↦(TM,gS), where the tangent bundle TM is equipped with the Sasaki metric gS. The constrained variational problem is studied, where variations are confined to vector fields, and the corresponding critical point condition characterizes biharmonic vector fields. We provide examples of biharmonic vector fields on three dimensional Lie groups equipped with a left-invariant Lorentzian metric and the Gödel universe.