Hypersurfaces of six-dimensional nearly Kähler manifolds

In the context of six-dimensional homogeneous nearly Kähler manifolds, we prove that $\mathbb S^6$ is the only ambient space admitting constant sectional curvature hypersurfaces. In order to do so, we prove first that in $\mathbb S^3\times\mathbb S^3$, $\mathbb C P^3$ and $F(\mathbb C^3)$, any hypersurface with constant sectional curvature is $η$-quasi umbilical, where $η$ is the dual one-form of the Reeb vector field. Then, we use the non-existence of such hypersurfaces in these spaces. Additionally, we characterize hypersurfaces of six-dimensional nearly Kähler manifolds which are Sasakian, nearly Sasakian, co-Kähler and nearly cosymplectic.