# Hypersurfaces of the homogeneous nearly K\"ahler $\mathbb{S}^6$ and $\mathbb{S}^3\times\mathbb{S}^3$ with anticommutative structure tensors

## TL;DR

This paper characterizes hypersurfaces in homogeneous nearly Kähler spheres, showing that certain anticommutative tensor conditions imply the hypersurface is totally geodesic or does not exist in specific cases.

## Contribution

It provides a complete characterization of hypersurfaces satisfying an anticommutative tensor condition in homogeneous nearly Kähler spheres, revealing their geometric properties.

## Key findings

- Hypersurfaces in $
abla$-Kähler $ ext{S}^6$ satisfying $A+ A=0$ are totally geodesic.
- No hypersurfaces in $
abla$-Kähler $ ext{S}^3 imes ext{S}^3$ satisfy $A+ A=0$.
- The condition $A+ A=0$ characterizes special geometric structures in these manifolds.

## Abstract

Each hypersurface of a nearly K\"ahler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $\phi$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $A\phi+\phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly K\"ahler manifold $\mathbb{S}^3\times\mathbb{S}^3$ does not admit a hypersurface that satisfies the condition $A\phi+\phi A=0$.