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

Abstract

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

14 pages. All comments are welcome