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We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was iniated by Pantilie, who shows that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space with an almost generalized complex structure. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperk\"ahler manifolds using the generalized Bismut connection, introduced by Gualtieri., Comment: 19 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

The aim of this article is to use generalized complex structures in order to extend the definition of twistor spaces given by Penrose. We will adapt the integrability result of Atiyah, Hitchin and Singer. We will deduce new correspondences betwenn differential geometry and (generalized) complex geometry. In the last section we will show how these results generalized Bredthauer's work., Comment: 19 pages, in French

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperk\"ahler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic K\"ahler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperk\"ahler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic K\"ahler manifold of non-positive scalar curvature. We check that no non-K\"ahler strong KT manifold occurs as such a twistor space.

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., Comment: 23 pages