Flat nearly Kähler manifolds

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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≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form ζ∈ Λ³ (Cᵐ)*. The first nontrivial example occurs in dimension 4m=12.