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

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

2005-03-08

arxiv.org/abs/math/0503150v3

Mathematics

Differential Geometry

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.

Scientific paper

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.

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