Invariant nearly-Kähler structures

Abstract

This paper considers invariant almost Hermitian structures on a flag manifold G / P = U / K where G is a complex semi-simple Lie group, P is a parabolic subgroup of G, U is a compact real form of G and K = U ∩ P is the centralizer of a torus. The main result shows that there are nearly-Kähler structures in G / P which are not Kähler if and only if G / P has height two. This proves for the flag manifolds a conjecture by Wolf and Gray.