A characterization of minimal Lagrangian submanifolds of the nearly Kähler $G \times G$

Abstract

We investigate Lagrangian submanifolds in the nearly Kähler manifold $G \times G$. First we review the construction of a nearly Kähler structure on the Lie group product $G \times G$, where $G$ is a Lie group with a bi-invariant metric. This construction was proposed by K. Sekigawa to E. Abbena and S. Garbiero. An example of this construction is the homogeneous nearly Kähler manifold $\mathbb{S}^{3}\times \mathbb{S}^{3}$, where $G=\mathbb{S}^{3}$ with its standard metric. It is known that this construction on $G \times G$ gives a nearly Kähler structure. To get our main result, we extend the notion of angle functions of a Lagrangian submanifold proposed by B. Dioos, L. Vrancken and X. Wang in the case of $\mathbb{S}^{3}\times \mathbb{S}^{3}$. These angle functions are useful to characterize minimal Lagrangian submanifolds in the NK manifold $G \times G$. We prove our main result: a Lagrangian submanifold is minimal if and only if the sum of its angle functions is constant. We give five examples of Lagrangian submanifolds: three canonical examples and other two examples using an element in the center of $G$.