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].