Hypersurfaces of the nearly Kähler twistor spaces $\mathbb{C}\mathrm{P}^3$ and ${\mathbb{F}_{1,2}}$

Abstract

In this article, we show that a hypersurface of the nearly Kähler ${\mathbb{C}\mathrm{P}}^3$ or ${\mathbb{F}_{1,2}}$ cannot have its shape operator and induced almost contact structure commute together. This settles the question for six-dimensional homogeneous nearly Kähler manifolds, as the cases of ${\mathbb{S}}^6$ and ${\mathbb{S}}^3 \times {\mathbb{S}}^3$ were previously solved, and provides a counterpart to the more classical question for the complex space forms ${\mathbb{C}\mathrm{P}}^n$ and ${\mathbb{C}\mathrm{H}}^n$. The proof relies heavily on the construction of ${\mathbb{C}\mathrm{P}}^3$ and ${\mathbb{F}_{1,2}}$ as twistor spaces of ${\mathbb{S}^{4}}$ and ${\mathbb{C}\mathrm{P}}^2$.