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

TL;DR

This paper classifies Lagrangian submanifolds in nearly Kähler ^3^3 based on their constant angle functions, revealing conditions for being totally geodesic or having constant sectional curvature, and identifying specific angle values.

Contribution

It provides a classification of Lagrangian submanifolds with constant or partially constant angle functions in nearly Kähler ^3^3, extending previous results and introducing new constructions.

Findings

Submanifolds with all constant angles are either totally geodesic or have constant sectional curvature.

If exactly one angle is constant, it must be 0, /3, or 2/3.

New classifications are obtained using specific constructions and previous classifications.

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.