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

Differential Geometry