KAHLER CURVATURE IDENTITIES FOR TWISTOR SPACES

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.

International Journal of Geometric Methods in Modern Physics, 2014

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on CP 3 is a non-Kähler Hermitian-Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein-Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z. 190 (1985) 39-43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom. 7 (2007) 207-214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

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-

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.

Osaka Journal of Mathematics, 2008

We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space Z of such a G-structure is endowed with a field of involutions J ∈ ¼(End T Z ) and a J -invariant distribution H Z . We study the conditions for the integrability of J and for the (para-)holomorphicity of H Z . Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces (Z¯, J , H Z ),¯= ±1, such that J 2 =¯Id. We prove that in both cases J is integrable (recovering results of Blair, Davidov and Muskarov) and that H Z defines a holomorphic (¯= −1) or para-holomorphic (¯= +1) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on Z¯.

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.

Annali di Matematica Pura ed Applicata, 2004

A class of minimal almost complex submanifolds of a Riemannian manifold M 4n with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifoldM 4n of non zero scalar curvature, in particular, whenM 4 is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M 2n of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M 2n ofM 4n is the projection of a holomorphic Legendrian submanifold L 2n ⊂ Z of the twistor space (Z, H) ofM 4n , considered as a complex contact manifold with the natural holomorphic contact structure H ⊂ T Z. Any Legendrian submanifold of the twistor space Z is defined by a generating holomorphic function. This is a natural generalization of Bryant's construction of superminimal surfaces in S 4 = H P 1 .

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.

Rocky Mountain Journal of Mathematics, 2005

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. . 53C28, 32L25, 53C26, 53C50.