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

Abstract

In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere ${\mathbb{S}}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of ${\mathbb{S}}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic ${\mathbb{S}}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.

12 pages. In this updated version we include a new rigidity theorem i.e. Theorem 4.1. All comments are welcome

References in corpus (1)

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

Cited by in corpus (1)

• A new characterization of the Calabi torus in the unit sphere