Flat nearly Kähler manifolds (Q2462630)

A nearly pseudo-Kähler manifold is a pseudo-Riemannian manifold \((M,g)\) with an skew-symmetric almost complex structure \(J\) such that \((\nabla_XJ)Y = (\nabla_YJ)X\) for \(X,Y \in TM\) where \(\nabla\) is the Levi-Civita connection. Such manifold admits a unique connection with skew-symmetric torsion which preserves \(g\) and \(J\). The manifold is called strict if \(\nabla J \neq 0\). The authors classify \(\nabla\)-flat strict nearly pseudo-Kähler manifolds. Any such manifold is locally a direct product of a flat pseudo-Kähler factor and a strict flat nearly pseudo-Kähler manifold of split signature \((2m,2m)\) with \(m \geq 3\). Moreover, the geometry of the second factor is encoded in a complex \(3\)-form \(\zeta \in \Lambda^3(\mathbb C^m)^*\). The first non-trivial example occurs in dimension \(4m =12\). The relations with \(tt^*\)-structures (topological-antitopological fusion structures) and duality between nearly pseudo-Kähler structures and special pseudo-Kähler structures are discussed.