Minimality and harmonicity for Hopf vector fields

Abstract

We determine when the Hopf vector fields on orientable real hypersurfaces $(M,g)$ in complex space forms are minimal or harmonic. Furthermore, we determine when these vector fields give rise to harmonic maps from $(M,g)$ to the unit tangent sphere bundle $(T_1M,g_S)$. In particular, we consider the special case of Hopf hypersurfaces and of ruled hypersurfaces. The Hopf vector fields on Hopf hypersurfaces with constant principal curvatures provide examples. The minimal ruled real hypersurfaces form another class of particular examples.