Hypersurfaces of six-dimensional nearly K\"ahler manifolds

Abstract

In the context of six-dimensional homogeneous nearly K\"ahler 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 $\eta$-quasi umbilical, where $\eta$ 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 K\"ahler manifolds which are Sasakian, nearly Sasakian, co-K\"ahler and nearly cosymplectic.