  We investigate Gauss maps associated to great circle fibrations of $S^3$. We
show that the associated Gauss map to such a fibration is harmonic
(respectively minimal) if and only if the unit vector field generating the
great circle foliation is harmonic (respectively minimal). These results can be
viewed as analogues of the classical theorem of Ruh and Vilms about the
harmonicity of the Gauss map of a minimal submanifold in the euclidean space.
Moreover, we prove that a harmonic or minimal unit vector field in $S^3$ with
great circle integral curves is a Hopf vector field.
