Homogeneous nearly K\"{a}hler manifolds
Abstract
We classify six-dimensional homogeneous nearly K\"{a}hler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly K\"{a}hler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are , the complex projective space , the flag manifold and the sphere . We develop, about each of these spaces, a distinct aspect of nearly K\"{a}hler geometry and make in the same time a sharp description of its specific homogeneous structure.
Cite
@article{arxiv.math/0612655,
title = {Homogeneous nearly K\"{a}hler manifolds},
author = {Jean-Baptiste Butruille},
journal= {arXiv preprint arXiv:math/0612655},
year = {2007}
}
Comments
This is the english version of an older article written in french (Classification des vari\'{e}t\'{e}s approximativement k\"{a}hleriennes homog\`{e}nes, 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 K\"{a}hler manifolds