  The bienergy of smooth maps between Riemannian manifolds, when restricted to
unit vector fields, yields two different variational problems depending on
whether one takes the full functional or just the vertical contribution. Their
critical points, called biharmonic unit vector fields and biharmonic unit
sections, form different sets. Working with surfaces, we first obtain general
characterizations of biharmonic unit vector fields and biharmonic unit sections
under conformal change of the metric. In the case of a 2-dimensional torus,
this leads to a proof that biharmonic unit sections are always harmonic and a
general existence theorem, in each homotopy class, for biharmonic unit vector
fields.
