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arXiv · 1902.01641

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

Zejun Hu·Jiabin Yin·Bangchao Yin
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Abstract

In this paper, we study Lagrangian submanifolds of the homogeneous nearly K\"ahler $6$-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.

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Zejun Hu, Jiabin Yin, Bangchao Yin. 2019-02-05. Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly K\"ahler $\mathbb{S}^6(1)$. https://doi.org/10.1016/j.geomphys.2019.06.003

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