Lagrangian Submanifolds with Constant Angle Functions of the nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$

Abstract

We study Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$ with respect to their, so called, angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follows from a recent paper of B. Dioos, L. Vrancken and X. Wang (arXiv:1604.05060). Moreover, we show that if precisely one angle function is constant, then it must be equal to $0,\frac{\pi}{3}$ or $\frac{2\pi}{3}$. Using then two remarkable constructions together with the classification of Lagrangian submanifolds of which the first component has nowhere maximal rank, we obtain a classification of such Lagrangian submanifolds.