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 en utilisant la distribution horizontale de la connexion hermitienne canonique. On montre qu’une condition nécessaire d’intégrabilité de est que la variété soit de type dans la classification de Gray et Hervella. Dans la deuxième partie on montre alors que les seules variétés de type 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 ou à une variété kählérienne, Bochner-plate.
We consider the reduced twistor space of an almost Hermitian manifold , after O’Brian and Rawnsley (Ann. Global Anal. Geom., 1985). We concentrate on dimension 6. This space has a natural almost complex structure associated with the canonical Hermitian connection. A necessary condition for the integrability of on is that the manifold belongs to the class of Gray, Hervella. In a second part, we then show that the almost Hermitian manifolds of type are all locally conformally nearly Kähler in dimension 6. Finally, is integrable if and only if is locally conformal to the sphere or to a Bochner-flat Kähler manifold.
Mot clés : géométrie presque hermitienne, espaces de twisteurs, structures ${\rm SU}(3)$
Keywords: almost Hermitian geometry, twistor spaces, ${\rm SU}(3)$-structures
Butruille, Jean-Baptiste 1
@article{AIF_2007__57_5_1451_0,
author = {Butruille, Jean-Baptiste},
title = {Espace de twisteurs d{\textquoteright}une vari\'et\'e presque hermitienne de dimension 6},
journal = {Annales de l'Institut Fourier},
pages = {1451--1485},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {57},
number = {5},
year = {2007},
doi = {10.5802/aif.2301},
mrnumber = {2364136},
zbl = {1130.53021},
language = {fr},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2301/}
}
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Butruille, Jean-Baptiste. Espace de twisteurs d’une variété presque hermitienne de dimension 6. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1451-1485. doi : 10.5802/aif.2301. https://aif.centre-mersenne.org/articles/10.5802/aif.2301/
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