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In this paper, we study Lagrangian submanifolds of the homogeneous nearlyK\"ahler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, wederive a Simons' type integral inequality in terms of the second fundamentalform for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, weshow that the equality sign occurs if and only if the Lagrangian submanifold iseither the totally geodesic $\mathbb{S}^3(1)$ or theDillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math SocJapan, 42: 565-584, 1990.

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