Espace de twisteurs d’une variété presque hermitienne de dimension 6  [ The twistor space of a dimensional almost Hermitian manifold ]
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, p. 1451-1485

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 W 4 of Gray, Hervella. In a second part, we then show that the almost Hermitian manifolds of type W 1 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.

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 W 4 dans la classification de Gray et Hervella. Dans la deuxième partie on montre alors que les seules variétés de type W 1 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.

DOI : https://doi.org/10.5802/aif.2301
Classification:  53C15,  53C28,  53C10
Keywords: almost Hermitian geometry, twistor spaces, SU (3)-structures
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     author = {Butruille, Jean-Baptiste},
     title = {Espace de twisteurs d'une vari\'et\'e presque hermitienne de dimension 6},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     pages = {1451-1485},
     doi = {10.5802/aif.2301},
     mrnumber = {2364136},
     zbl = {1130.53021},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2007__57_5_1451_0}
}
Butruille, Jean-Baptiste. Espace de twisteurs d’une variété presque hermitienne de dimension 6. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1451-1485. doi : 10.5802/aif.2301. http://www.numdam.org/item/AIF_2007__57_5_1451_0/

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