Espace de twisteurs d'une variete presque hermitienne de dimension 6

Abstract

We consider the reduced twistor space $Z$ of an almost Hermitian manifold $M$, after O'Brian and Rawnsley (Ann. Global Anal. Geom., 1985). We concentrate on dimension 6. This space has a natural almost complex structure $\mathcal J$ associated to the canonical Hermitian connection. A necessary condition for the integrability of $\mathcal J$ on $Z$ is that the manifold belongs to the class $W_1 \oplus W_4$ of Gray, Hervella. In a second part, we then show that the almost Hermitian manifolds of type $W_1 \oplus W_4$ are all locally conformally nearly K\"{a}hler in dimension 6. Finally, $\mathcal J$ is integrable if and only if $M$ is locally conformal to the sphere $S^6$ or to a Bochner-flat K\"{a}hler manifold.