On surfaces whose twistor lifts are harmonic sections (Q933664)

The author studies oriented surfaces in \(4\)-dimensional oriented Riemannian manifolds, whose twistor lifts are harmonic sections. These surfaces are characterized in terms of their fundamental forms. Under certain assumptions for the curvature tensor, as a corollary, it is proved that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. In the case when the ambient space is Euclidean, a lower bound for the vertical energy of the twistor lifts is provided. Further, the author characterizes the oriented compact connected surfaces of \({\mathbb R}^4\) which obey supplementary assumptions on the mean curvature vector field, in the case when the twistor lift is a harmonic section and its vertical energy density is a constant.