Twistor space of a generalized quaternionic manifold

2005

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.

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¯.

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.

Journal Fur Die Reine Und Angewandte Mathematik, 2004

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.

New Developments in Differential Geometry, 1996

On an almost quaternionic manifold (M 4n , Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n ≥ 2, we prove that the existence of two compatible complex structures I 1 , I 2 = ±I 1 forces (M 4n , Q) to be quaternionic. If n = 1, that is (M 4 , Q) = (M 4 , [g], or) is an oriented conformal 4-manifold, we prove a maximum principle for the angle function I 1 , I 2 of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M 4n , Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.

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 .

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.

Transactions of the American Mathematical Society, 1999

On an almost quaternionic manifold (M 4n , Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n ≥ 2, we prove that the existence of two compatible complex structures I 1 , I 2 = ±I 1 forces (M 4n , Q) to be quaternionic. If n = 1, that is (M 4 , Q) = (M 4 , [g], or) is an oriented conformal 4-manifold, we prove a maximum principle for the angle function I 1 , I 2 of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M 4n , Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.