Search

In 2002 Meersseman-Verjovsky [2] constructed a smooth, codimension-one, foliation on 5-sphere by complex surfaces with two compact leaves. The aim of this note is to improve their construction in order to give a smooth foliation on 5-sphere by complex surfaces with only one compact leaf.

Let (M,g) be a Riemannian 4-manifold. The twistor space Z->M is a CP1-bundle whose total space Z admits a natural metric h. The aim of this article is to study properties of complex structures on (Z,h) which are compatible with the CP1-fibration and the metric h. The results obtained enable us to translate some metric properties on M in terms of complex properties on its twistor space Z., 23 pages

To give an almost quaternionic structure on a 4n-manifold $M$ is equivalent to give its bundle of twistors $Z(Q)\longrightarrow M$. When $Q$ is invariant under a torsion free connection, $Z(Q) $ can be provided with an almost complex structure $ \mathbb J $. In the case $ n = 1 $ Atiyah, Hitchin and Singer have related the integrability of $ \mathbb J $ to the geometry of $ (M, Q) $. For $ n> 1 $ Salamon showed that the almost complex structure $ \mathbb J $ on $ Z (Q) $ is always integrable. The purpose of this article is to extend these results to the generalized complex geometry. We begin by defining the concept of almost generalized quaternionic manifolds $ (M, g, \mathcal Q ) $. We will see that we can associate a twistor space denoted by $ \mathcal Z( \mathcal Q) $ which is a $ \mathbb S^2$-bundle over $ M $. When $\mathcal Q$ is invariant under a generalized torsion free connection, then $ \mathcal Z(\mathcal Q) $ comes with an almost generalized complex structure $\mathbb J$. Whatever the dimension of $ M $ is, we give a criterion for integrability of the almost generalized complex structure $ \mathbb J $ on $ \mathcal Z(\mathcal Q) $. In the particular case where $ (M, g,\mathcal Q) $ is a generalized quaternionic K\"ahler manifold, we show that $\mathbb J$ is always integrable as soon as $n>1$. We illustrate this work by giving several examples of generalized quaternionic K\"ahler manifolds for which the almost generalized complex structure $ \mathbb J $ on the twistor space $ \mathcal Z(\mathcal Q) $ is integrable., Comment: 23 pages, article in French

To give an almost quaternionic structure on a 4n-manifold $M$ is equivalent to give its bundle of twistors $Z(Q)\longrightarrow M$. When $Q$ is invariant under a torsion free connection, $Z(Q) $ can be provided with an almost complex structure $ \mathbb J $. In the case $ n = 1 $ Atiyah, Hitchin and Singer have related the integrability of $ \mathbb J $ to the geometry of $ (M, Q) $. For $ n> 1 $ Salamon showed that the almost complex structure $ \mathbb J $ on $ Z (Q) $ is always integrable. The purpose of this article is to extend these results to the generalized complex geometry. We begin by defining the concept of almost generalized quaternionic manifolds $ (M, g, \mathcal Q ) $. We will see that we can associate a twistor space denoted by $ \mathcal Z( \mathcal Q) $ which is a $ \mathbb S^2$-bundle over $ M $. When $\mathcal Q$ is invariant under a generalized torsion free connection, then $ \mathcal Z(\mathcal Q) $ comes with an almost generalized complex structure $\mathbb J$. Whatever the dimension of $ M $ is, we give a criterion for integrability of the almost generalized complex structure $ \mathbb J $ on $ \mathcal Z(\mathcal Q) $. In the particular case where $ (M, g,\mathcal Q) $ is a generalized quaternionic K��hler manifold, we show that $\mathbb J$ is always integrable as soon as $n>1$. We illustrate this work by giving several examples of generalized quaternionic K��hler manifolds for which the almost generalized complex structure $ \mathbb J $ on the twistor space $ \mathcal Z(\mathcal Q) $ is integrable., 23 pages, article in French

In this paper we use the twistor theory in order to build non standard complex structures (with a meaning which we define) on products of 4-manifolds with the sphere of dimension two. To that end, we enumerate the set of complex surfaces whose twistor space is topologically trivial. Among these surfaces, we determine those which admit a spin structure or an anti-selfdual riemannian metric.