Hypersurfaces of six-dimensional nearly Kähler manifolds

Abstract

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 $\eta$-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 Kähler manifolds which are Sasakian, nearly Sasakian, co-Kähler and nearly cosymplectic.

17 pages, 8 tables