Tohoku Mathematical Journal
2021
December
SECOND SERIES VOL. 73, NO. 4

Tohoku Math. J.
73 (2021), 627-642

Title HYPERSURFACES OF THE NEARLY KÄHLER TWISTOR SPACES $\mathbb{C}\mathrm{P}^3$ AND ${\mathbb{F}_{1,2}}$

Author Guillaume Deschamps and Eric Loubeau

(Received March 3, 2020, revised September 9, 2020)
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 ${\mathbb{C}\mathrm{P}}^2$.

Mathematics Subject Classification. Primary 32L25; Secondary 53C28, 53C55, 53C15.
Key words and phrases. Nearly Kahler manifolds, submanifolds, twistor theory.

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