Almost-complex and almost-product Einstein manifolds from a variational principle

It is shown that the first-order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterization of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered.