Hypersurfaces of the homogeneous nearly Kahler $mathbb{S}^6$ and $mathbb{S}^3timesmathbb{S}^3$ with anticommutative structure tensors

Each hypersurface of a nearly Kahler 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 $Aphi+phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kahler manifold $mathbb{S}^3timesmathbb{S}^3$ does not admit a hypersurface that satisfies the condition $Aphi+phi A=0$.