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Variétés de Gray et géométries spéciales en dimension 6

Abstract : We consider 6-dimensional almost Hermitian manifolds with a further reduction to SU(3) induced by the differential of the Kähler form dω. We use the fact stressed by Hitchin that two differential forms – the Kähler form ω and a complex volume 3-form Ψ – are enough to define an SU(3)-structure on a 6-manifold. Moreover, by a result of Chiossi, Salamon, the differential of these forms determine the 1-jet of the SU(3)-structure. An important example is given by the nearly Kähler, non Kählerian manifolds in dimension 6. Here the reduction is global. We classify the homogeneous examples and, thanks to previous results of Cleyton, Swann and Nagy, solve positively the conjecture of Gray and Wolf that all strictly nearly Kähler homogeneous manifolds are 3-symmetric. Another result concerns a submanifold of the twistor space, called the “reduced twistor space” of an almost Hermitian manifold. This space is equipped with a natural almost complex structure which we show is integrable if and only if the manifold is locally conformal to a Bochner-flat Kähler manifold or to the sphere S6. Furthermore, in the course of the proof, we obtain the following reduction of the Gray-Hervella classification : every almost Hermitian manifold of type W1+W4 is locally conformally nearly Kähler in dimension 6.
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Contributor : Jean-Baptiste Butruille <>
Submitted on : Thursday, December 7, 2006 - 10:58:12 AM
Last modification on : Wednesday, March 27, 2019 - 4:10:21 PM
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  • HAL Id : tel-00118939, version 1



Jean-Baptiste Butruille. Variétés de Gray et géométries spéciales en dimension 6. Mathématiques [math]. Ecole Polytechnique X, 2005. Français. ⟨tel-00118939⟩