Homogeneous nearly Kähler manifolds
Jean-Baptiste Butruille
TL;DR
This work resolves Gray and Wolf's conjecture in dimension six by proving that every homogeneous nearly Kähler 6-manifold is a Riemannian 3-symmetric space with its canonical almost Hermitian structure, and classifies all such spaces. It identifies exactly four 6D examples—$S^3\times S^3$, $S^6$, $\mathbb{C}P^3$, and $\mathbb F^3$—each carrying a unique invariant nearly Kähler structure up to homothety, and it develops invariant $SU(3)$-structures via differential forms, SU(3) reductions, and twistor-theoretic perspectives. The paper analyzes left-invariant structures on $S^3\times S^3$ through a cyclic co-frame, relates $\mathbb{C}P^3$ and $\mathbb F^3$ to twistor spaces, and links $S^6$ to weak holonomy and $G_2$-cones, synthesizing these viewpoints with a Lie-theoretic classification of 3-symmetric spaces. This comprehensive synthesis confirms the 6D homogeneous NK landscape, clarifies each example’s geometry, and situates the result within Nagy’s higher-dimensional program for special geometries with torsion.
Abstract
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 $S^3 \times S^3$, 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 Kähler geometry and make in the same time a sharp description of its specific homogeneous structure.