Killing vector fields and Lagrangian submanifolds of the nearly Kaehler \(S^6\) (Q1267275)

It is well known that the algebraic structure of the Cayley numbers can be used to define an almost complex structure \(J\) on the 6-dimensional unit sphere \(S^6\) turning \(S^6\) into a nearly Kähler manifold. A submanifold \(M\) of \(S^6\) is said to be totally real if \(J\) maps the tangent bundle \(TM\) of \(M\) into the normal bundle of \(M\), and it said to be almost complex if \(J\) leaves \(TM\) invariant. The author gives a local classification of all 3-dimensional totally real submanifolds \(M\) of \(S^6\) admitting a unit Killing vector field whose integral curves are great circles. Roughly, they can be constructed from either superminimal almost complex curves in \(S^6\) or via Hopf lifts from holomorphic curves in complex projective plane \(\mathbb{C} P^2\).