Ruled Lagrangian Submanifolds of the 6-Sphere

TL;DR

This paper explicitly describes ruled Lagrangian submanifolds in the nearly Kähler 6-sphere using Weierstrass formulas and classifies all such submanifolds satisfying natural second order conditions.

Contribution

It provides an explicit characterization of ruled Lagrangians in the 6-sphere and unifies known examples under a common geometric framework.

Findings

Explicit description of ruled Lagrangians using Weierstrass formulas

Classification of all second order families of Lagrangians in the 6-sphere

Identification of all known examples as ruled by circles of constant radius

Abstract

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