Hypersurfaces of six-dimensional nearly K\"ahler manifolds

TL;DR

This paper classifies hypersurfaces with constant sectional curvature in six-dimensional nearly K"ahler manifolds, showing they only exist in the sphere and characterizing various special hypersurfaces in these spaces.

Contribution

It proves that only the 6-sphere admits constant sectional curvature hypersurfaces among homogeneous nearly K"ahler manifolds and characterizes special hypersurfaces in these spaces.

Findings

Only $ extbf{S}^6$ admits constant sectional curvature hypersurfaces.

Hypersurfaces with constant sectional curvature in certain spaces are $ extbf{ exteta}$-quasi umbilical.

Characterization of hypersurfaces that are Sasakian, nearly Sasakian, co-K"ahler, or nearly cosymplectic.

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.