Homogeneous nearly Kähler manifolds

Details Homogeneous nearly Kähler manifolds Homogeneous nearly Kähler manifolds

2006-12-21

arxiv.org/abs/math/0612655v1

Mathematics

Differential Geometry

This is the english version of an older article written in french (Classification des vari\'{e}t\'{e}s approximativement k\"{a

Scientific paper

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 $S^3 \times S^3$, the complex projective space $\CM P^3$, the flag manifold $\mathbb F^3$ and the sphere $S^6$. 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.

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