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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 $CP^2$
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In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly Kahler) manifold $mathbf{S}^3timesmathbf{S}^3$. First, we show that any Hopf hypersurface of the homogeneous NK $mathbf{S}^3timesmathbf{S}^3$ does not admit two distinct principal curvatures. Then, for the important class of Hopf hypersurfaces with three distinct principal curvatures, we establish a complete classification under the additional condition that their holomorphic distributions ${U}^perp$ are preserved by the almost product structure $P$ of the homogeneous NK $mathbf{S}^3timesmathbf{S}^3$.
We formulate the deformation theory for instantons on nearly Kahler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly Kahler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3).
Each hypersurface of a nearly Kahler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $phi$. In this paper, we show that, in the homogeneous NK $mathbb{S}^6$ a hypersurface satisfies the condition $Aphi+phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kahler manifold $mathbb{S}^3timesmathbb{S}^3$ does not admit a hypersurface that satisfies the condition $Aphi+phi A=0$.
We study the local geometry of 4-manifolds equipped with a emph{para-Kahler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-vanishing Einstein constant and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartans method of equivalence to produce a large number of explicit examples of pKE metrics with nonvanishing Einstein constant whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type $D,$ we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding emph{Cartan connections} satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
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