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

## TL;DR

This paper establishes rigidity theorems for Lagrangian submanifolds in the nearly Kähler 6-sphere, deriving integral inequalities and characterizing cases of equality as totally geodesic spheres or Berger spheres.

## Contribution

It introduces a Simons' type integral inequality for compact Lagrangian submanifolds in the nearly Kähler 6-sphere and characterizes the equality cases.

## Key findings

- Derived a Simons' type integral inequality involving the second fundamental form.
- Characterized the equality case as either totally geodesic or Berger spheres.
- Provided rigidity results for Lagrangian submanifolds in the nearly Kähler 6-sphere.

## 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.