Espace de twisteurs d’une variété presque hermitienne de dimension 6

Butruille, Jean-Baptiste

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1451-1485.

On s’intéresse à l’espace de twisteurs réduit d’une variété presque hermitienne, en relisant un article de N.R.O’Brian et J.H.Rawnsley (Ann. Global Anal. Geom., 1985). On traite la question laissée ouverte de la dimension 6. Cet espace est muni d’une structure presque complexe $𝒥$ en utilisant la distribution horizontale de la connexion hermitienne canonique. On montre qu’une condition nécessaire d’intégrabilité de $𝒥$ est que la variété soit de type $W_1\oplus W_4$ dans la classification de Gray et Hervella. Dans la deuxième partie on montre alors que les seules variétés de type $W_1\oplus W_4$ en dimension 6 sont les variétés localement conformément « nearly Kähler ». Finalement la structure presque complexe de l’espace de twisteurs réduit est intégrable si et seulement si la variété est localement conforme à la sphère $S^6$ ou à une variété kählérienne, Bochner-plate.

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 $𝒥$ associated with the canonical Hermitian connection. A necessary condition for the integrability of $𝒥$ 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, $𝒥$ is integrable if and only if $M$ is locally conformal to the sphere $S^6$ or to a Bochner-flat Kähler manifold.

Classification : 53C15, 53C28, 53C10
Mot clés : géométrie presque hermitienne, espaces de twisteurs, structures ${\rm SU}(3)$
Keywords: almost Hermitian geometry, twistor spaces, ${\rm SU}(3)$-structures

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