A note on para-holomorphic Riemannian–Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kahler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kahler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kahlerian–Norden Einstein metric.