Johann Davidov

TWISTOR SPACES WITH HERMITIAN RICCI TENSOR

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kahler manifold with Hermitian Ricci tensor is Kahlerian.

Hyperbolic Twistor Spaces

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their holomorphic functions.

Twistorial construction of generalized Kähler manifolds

Abstract The twistor method is applied for obtaining examples of generalized Kahler structures which are not yielded by Kahler structures.

Twistor spaces of generalized complex structures

Abstract The twistor construction is applied for obtaining examples of generalized complex structures (in the sense of Hitchin) that are not induced by a complex or a symplectic structure.

Compact self-dual Hermitian surfaces

In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kahler.

Generalized Pseudo-Kähler Structures

In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.

Harmonic almost-complex structures on twistor spaces

We prove that the Atiyah-Hitchin-Singer [1] and Eells-Salamon [6] almost-complex structures on the negative twistor space of an oriented Riemannian four-manifold are harmonic in the sense of C. Wood [17, 18] if and only if the base manifold is, respectively, self-dual or self-dual and of constant scalar curvature. The stability of these almost-complex structures is also discussed.

Curvature Properties of the Chern Connection of Twistor Spaces

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h_t compatible with the almost complex structures J_1 and J_2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z,h_t,J_n), n=1,2 and find the geometric conditions on M under which the curvature of its Chern connection D^n is of type (1,1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D^n and show that the Nijenhuis tensor of J_2 is D^2-parallel provided the base manifold M is Einstein and self-dual.

Twistorial Examples of ∗-Einstein Manifolds

In this paper we study the twistor spaces of oriented Riemannianfour-manifolds as a source of almost-Hermitian *-Einstein manifoldsand show that some results in dimension four related to the RiemannianGoldberg–Sachs theorem cannot be extended to higher dimensions.

Twistorial construction of minimal hypersurfaces

Every almost Hermitian structure