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

Gauss maps of harmonic and minimal great circle fibrations