math/0612655Dec 21, 2006This is the english version of an older article written in french (Classification des variétés approximativement kähleriennes homogènes, Ann. Global Anal. Geom. 27, 201-225, 2005). It contains no new results. However, we modified the structure of the paper, simplified some proofs and added a lot of explanations, especially on 3-symmetric spaces. It can be read as a sort of survey on nearly Kähler manifolds
We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are S3×S3, the complex projective space P³, the flag manifold F3 and the sphere S6
Nearby in the stack
. We develop, about each of these spaces, a distinct aspect of nearly Kähler geometry and make in the same time a sharp description of its specific homogeneous structure.