Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly Kähler [formula omitted

Abstract

In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler 6-dimensional unit sphere S6(1). As the main result, we derive a Simons’ type integral inequality in terms of the second fundamental form of compact Lagrangian submanifolds of S6(1). Moreover, we show that this inequality is optimal and the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic S3(1) or the Dillen–Verstraelen–Vrancken’s Berger sphere S3 described in Dillen et al. (1990).