Holomorphic curvatures of twistor spaces

Proceeding of the Bulgarian Academy of Sciences, 2013

In this paper we provide twistorial examples of compact Hermitian manifolds with positive holomorphic bisectional curvature. We also observe that the so-called "squashed" metric on CP 3 , the twistor space of the sphere S 4 , is a non-Kähler Hermitian-Einstein metric of positive holomorphic bisectional curvature, thus showing that a recent result of Kalafat and Koca in complex dimension two cannot be extended in higher dimensions.

Journal of Geometry, 1998

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

Journal of Geometry, 1998

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

Illinois journal of mathematics

Acta Mathematica Hungarica, 1991

Coordinate-free formulas for the sectional curvatures of a family of pseudo-Riemannian metrics on a twistor space are obtained in terms of the curvature of the base four-manifold. The respective Ricci curvatures and the holoraorphic sectional curvatures of two natural almost complex structures are also discussed. . §1. Introduction. The twistor space of an oriented Riemannian 4-manlfold H Is the 2-sphere bundle Z on K consisting of the unit (-1)-eigenvectors of the Hodge star operator acting on A ^M. The 6-manif©ld Z admits a natural 1-parameter family of pseudo-RiemaimiaQi metrics 1^ , t ^ 0 . For t y 0, these metrics are definite and have been studied by ?riedrich & Kurke [ 4 ] in connection with the classification of self-dual Einstein 4-manlfolds with positive aacalar curvature . In [ 3 ] t Frtedrldi & Grunewald have given, the geometric conditions on M ensuring that 1^ , t?0 , is an Einstein metric • In the case t < 0 , h^ is indefinite and has been studied Tsjr Titter in [10] where local formulaB for the curvature and Rlcci forms have been obtained. K.Seklgawa. [8 j has considered the metrics iu , t > 0 , on the twistor space of am oriented Riemannian 2n-manifold» She main purpose of this paper is to give a coordinate-free formula; for the sectional curvature of the pseudo-Riemannian manifold (Z t h^.) in' terms of the curvature of H . This is3 achieved by/ means of the O'Keill formulas f 6 J for Riemannian submersions . As applications we discuss the Rlcci curvature of (Z, b^) and the holomorpMc sectional curvatures with respect to the almost complex structures on Z introduced.respectively by A ti yah, Hit chin & Singer [11 and Eells &. Salamon [2] . §2. Preliminaries, Let M be an oriented Rieiaannian 4-manifold with metric g . Then g induces a metric on the bundle of 2-vectors A ^M by the formula -2-

Proceedings of the American Mathematical Society, 1990

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-Kähler manifold with Hermitian Ricci tensor is Kählerian.

Journal of Geometry and Physics, 2004

In this paper we study two natural indefinite almost Hermitian structures on the hyperbolic twistor space of a four-manifold endowed with a neutral metric. We show that only one of these structures can be isotropic Kähler and obtain the precise geometric conditions on the base manifold ensuring this property.

Mathematische Zeitschrift, 2020

We study Hermitian metrics whose Bismut connection ∇ B satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition R B (x, y, z, w) = R B (Jx, Jy, z, w), for every tangent vectors x, y, z, w, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in [4, 30, 28]. Using the characterization of SKT almost abelian Lie groups in [5], we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.

Proceedings of the Steklov Institute of Mathematics, 2020

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.

We show that a compact complex surface together with an Einstein-Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling. This result relaxes the K\&quot;ahler condition in Berger&#39;s theorem, and the positivity condition on sectional curvature in a theorem proved by Koca. The techniques used in the proof are completely different from theirs.