Gauss maps of harmonic and minimal great circle fibrations

TL;DR

This paper explores the properties of Gauss maps related to great circle fibrations of the 3-sphere, establishing conditions under which these maps are harmonic or minimal, and characterizing special vector fields as Hopf fields.

Contribution

It demonstrates that the harmonicity or minimality of Gauss maps corresponds to that of the generating vector fields, and characterizes such fields as Hopf vector fields.

Findings

Gauss map is harmonic if and only if the vector field is harmonic.

Gauss map is minimal if and only if the vector field is minimal.

Harmonic or minimal vector fields with great circle integral curves are Hopf vector fields.

Abstract

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.