Hypersurfaces of nearly Kahler twistor spaces

TL;DR

This paper proves that hypersurfaces in certain six-dimensional nearly Kahler manifolds, specifically $CP^3$ and $F_{1,2}$, cannot have their shape operator and induced almost contact structure commute, extending known results to these spaces.

Contribution

It establishes a non-commutativity result for hypersurfaces in nearly Kahler $CP^3$ and $F_{1,2}$, filling a gap in the understanding of hypersurfaces in these manifolds.

Findings

Hypersurfaces in $CP^3$ and $F_{1,2}$ cannot have commuting shape operator and almost contact structure.

The result completes the classification for six-dimensional homogeneous nearly Kahler manifolds.

The proof uses the twistor space construction of these manifolds.

Abstract

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 $CP^2$