On isotropic Lagrangian submanifolds in the homogeneous nearly Kähler ${\mathbb{S}}^3\times {\mathbb{S}}^3$

Abstract

In this paper, we show that isotropic Lagrangian submanifolds in a $6$-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the $J$-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler ${\mathbb{S}}^3\times {\mathbb{S}}^3$ is also obtained. Here, a Lagrangian submanifold is called $J$-isotropic, if there exists a function $\lambda$, such that $g((\nabla h)(v,v,v),Jv)=\lambda$ holds for all unit tangent vector $v$.

18 pages. This article has been accepted for publication in SCIENCE CHINA Mathematics

References in corpus (1)

• Generalized Killing spinors and Lagrangian graphs

Cited by in corpus (1)

• Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly Kähler ${\mathbb{S}}^6(1)$