Hypersurfaces of nearly Kahler twistor spaces

Abstract

In this article, we show that a hypersurface of the nearly Kahler $C{P}^3$ 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 $C{P}^n$ and $C{H}^n$. The proof relies heavily on the construction of $C{P}^3$ and $F_{1,2}$ as twistor spaces of $S^4$ and $C{P}^2$

arXiv commentFinal version, to appear in Tohoku Mathematical Journal