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 characterisation of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered. On leave from the Institute of Theoretical Physics, University of Wroc law, pl. Maksa Borna 9, 50-204 WROC LAW (POLAND), E-mail: [email protected] Permanent address: Steklov Mathematical Institute, Russian Academy of Sciences, Vavilov St. 42, GSP–1, 117966 MOSCOW (RUSSIA).