Gauss maps of harmonic and minimal great circle fibrations

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere \(\mathbb {S}^3\). 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 on \(\mathbb {S}^3\), whose integral curves are great circles, is a Hopf vector field.