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Johann Davidov

Publications and source records attributed to Johann Davidov.

At least 19 recordsLinked to original sources
arXiv2026

Gray-Hervella classes on product twistor spaces

Motivated by generalized geometry (in the sense of Hitchin), the product bundle ${\mathcal Z}\times_{M} {\mathcal Z}$ of the twistor space ${\mathcal Z}$ of a Riemannian manifold $(M,g)$ is considered. The product twistor space admits a natural family of Riemannian metrics and four compatible almost complex structures, analogs of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space. The Gray-Hervellal classes of these almost Hermitian structures are determined in the case when the dimension of the base manifold $M$ is four.

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arXiv2024

On a natural map between twistor spaces

A diffeomorphism between the twistor spaces of two Riemannian metrics on a smooth manifold preserving the fibres is defined based on a well-known construction. It is shown that this bundle isomorphim is a holomorphic map with respect to the Atiyah-Hitchin-Singer, respectively Eells-Salamon, almost complex structure if and only if the two metrics are conformal, respectively homothetic. In these cases, the diffeomorphism is the identity map and the result obtained provides an interpretation of the well-known fact that the Atiyah-Hitchin-Singer almost complex structure of a twistor space is invariant under conformal changes of the metric on the base manifold, while that of Eells-Salamon is not invariant in general. The more general problem of when an arbitrary bundle isomorphism between twisor spaces is holomorphic is also considered. Another problem discussed in the paper is when the diffeomorphism mentioned above is a harmonic map with respect to natural families of Riemannian metrics on the twistor spaces defined by means of the two Riemannian metrics. It is proved that if the metrics are conformal this happens if and only if they are homothetic.

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arXiv2022

Pseudo-harmonic Hermitian structures on Weyl manifolds

We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.

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arXiv2022

Complex surfaces and null conformal Killing vector fields

We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of signature (2,2). We establish first the topological types of pseudo-Hermitian surfaces admitting a nowhere vanishing null vector field. Then we show that a pair of orthogonal, pointwise linearly independent, null, conformal Killing vector fields defines a para-hyperhermitian structure and use this fact for a classification of the smooth compact four-manifolds admitting such a pair of vector fields. We also provide examples of neutral metrics with two orthogonal, pointwise linearly independent, null Killing vector fields on most of these manifolds.

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arXiv2021

Harmonic Hermitian structures on Riemannian manifolds with skew-torsion

We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space considered with a natural family of Riemannian metrics defined by means of the metric and the given connection on the base manifold.

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arXiv2021

Curvature properties of twistor spaces

In this paper we review some results on the Riemannian and almost Hermitian geometry of twistor spaces of oriented Riemannian $4$-manifolds with emphasis on their curvature properties.

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arXiv2019

Product twistor spaces and Weyl geometry

Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symmetric torsion. These structures are defined by means of the connection $D$ and four (Kähler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry and this is used to supply examples satisfying the conditions.

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arXiv2018

Generalized metrics and generalized twistor spaces

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry à la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms.

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arXiv2017

Almost complex structures that are harmonic maps

We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.

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arXiv2016

Harmonic almost Hermitian structures

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, a special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannain four-manifold.

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arXiv2016

Contact twistor spaces and almost contact metric structures

The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost $CR$-structure on its twistor space and Vezzoni has found the integrability condition for this structure. In the present paper it is observed that the $CR$-structure is induced by an almost contact metric structure. The main goal of the paper is to obtain necessary and sufficient conditions for normality of this structure in terms of the curvature of the given contact connection. Illustrating examples are discussed at the end of the paper.

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arXiv2014

Twistorial construction of minimal hypersurfaces

Every almost Hermitian structure $(g,J)$ on a four-manifold $M$ determines a hypersurface $Σ_J$ in the (positive) twistor space of $(M,g)$ consisting of the complex structures anti-commuting with $J$. In this note we find the conditions under which $Σ_J$ is minimal with respect to a natural Riemannian metric on the twistor space in the cases when $J$ is integrable or symplectic. Several examples illustrating the obtained results are also discussed.

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arXiv2014

Normality of the twistor space of a $5$-manifold with a $SO(3)$-structure

A manifold with an irreducible $SO(3)$-structure is a $5$-manifold $M$ whose structure group can be reduced to the group $SO(3)$, non-standardly imbedded in $SO(5)$. The study of such manifolds has been initiated by M. Bobieński and P. Nurowski who, in particular, have shown that one can define four $CR$-structures on a twistor-like $7$-dimensional space associated to $M$. In the present paper it is observed that these $CR$-structures are induced by almost contact metric structures. The purpose of the paper is to study the problem of normality of these structures. The main result gives necessary and sufficient condition for normality in geometric terms of the base manifold $M$. Examples illustrating this result are presented at the end of the paper.

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arXiv2012

Compact complex surfaces with geometric structures related to split quaternions

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that a compact oriented 4-manifold carries a para-hyperkähler structure iff it has a metric of split signature together with two parallel, orthogonal and null vector fields. Every compact complex surface admiting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperkähler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type $S^0$ and provide a list of compact complex surfaces which could carry para-hypercomplex structures.

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