Totally Real Minimal Surfaces with Non‐Circular Ellipse of Curvature in the Nearly Kähler S 6

In [ 2 ] we discussed almost complex curves in the nearly Kähler S 6 . These are surfaces with constant Kähler angle 0 or π and, as a consequence of this, are also minimal and have circular ellipse of curvature. We also considered minimal immersions with constant Kähler angle not equal to 0 or π, but with ellipse of curvature a circle. We showed that these are linearly full in a totally geodesic S 5 in S 6 and that (in the simply connected case) each belongs to the S 1 ‐family of horizontal lifts of a totally real (non‐totally geodesic) minimal surface in C P 2 . Indeed, every element of such an S 1 ‐family has constant Kähler angle and in each family all constant Kähler angles occur. In particular, every minimal immersion with constant Kähler angle and ellipse of curvature a circle is obtained by rotating an almost complex curve which is linearly full in a totally geodesic S 5 .