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 Kähler 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 Kähler manifold.

In french. Version 2. The paper has been abbreviated from 38 pages to 30. The proof of the actual lemma 5.2 has been precised. Three references were added : [3], [7] and mostly [9] (Cleyton, Ivanov, math.DG/0607487 : the authors completed our theorem 1 by proving that a 6-dimensional almost Hermitian manifold is locally conformally nearly Kähler if and only if it is globally conformally nearly Kähler)

Cited by in corpus (1)

• Conformal equivalence between certain geometries in dimension 6 and 7