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

It is shown that the first-order (Palatini) variational principle for a gen eric nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricc i square invariant leads to an almost-product Einstein structure or to an a lmost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kahl er 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-Kahler Einstein manifolds and almost-product Einstein manifolds is ob tained. Examples of such manifolds are considered. (C) 1999 American Instit ute of Physics. [S0022-2488(99)03107-2].