Ruled Lagrangian submanifolds of the 6-sphere

Abstract

This article sets out to serve a dual purpose. On the one hand, we give an explicit description of the Lagrangians in the nearly Kähler 6-sphere S 6 \mathcal {S}^6 which are ruled by circles of constant radius using ‘Weierstrass formulae’. On the other hand, we recognise all known examples of these Lagrangians as being ruled by such circles. Therefore, we describe all families of Lagrangians in S 6 \mathcal {S}^6 whose second fundamental form satisfies natural pointwise conditions—the so-called ‘second order families’.