  We will investigate the local geometry of the surfaces in the $7$-dimensional
Euclidean space associated to harmonic maps from a Riemann surface $\Sigma$
into $S^6$. By applying methods based on the use of harmonic sequences, we will
characterize the conformal harmonic immersions $\varphi:\Sigma\to S^6$ whose
associated immersions $F:\Sigma\to \mathbb{R}^7$ belong to certain remarkable
classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with
parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic
surfaces.
