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

Butruille, Jean-Baptiste

Annales de l'Institut Fourier, Tome 57 (2007), p. 1451-1485 / Harvested from Numdam

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Abstract

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.

Publié le : 2007-01-01

MR 2364136 | Zbl 1130.53021

DOI : https://doi.org/10.5802/aif.2301
Classification: 53C15, 53C28, 53C10
Mots clés: géométrie presque hermitienne, espaces de twisteurs, structures

SU (3)

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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},
     volume = {57},
     year = {2007},
     pages = {1451-1485},
     doi = {10.5802/aif.2301},
     zbl = {1130.53021},
     mrnumber = {2364136},
     language = {fr},
     url = {http://dml.mathdoc.fr/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, Tome 57 (2007) pp. 1451-1485. doi : 10.5802/aif.2301. http://gdmltest.u-ga.fr/item/AIF_2007__57_5_1451_0/

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