Flat nearly Kähler manifolds

Abstract

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature $(2m,2m)$ with $m\ge 3$. Moreover, the geometry of the second factor is encoded in a complex three-form $ζ\in Λ^3 (\mathbb{C}^m)^*$. The first nontrivial example occurs in dimension $4m=12$.