A note on para-holomorphic Riemannian Einstein manifolds

TL;DR

This paper investigates Einstein conditions for para-holomorphic Riemannian metrics within para-complex geometry, establishing connections with para-Kähler Norden metrics and demonstrating Einstein metrics on semi-simple para-complex Lie groups.

Contribution

It introduces a study of Einstein conditions for para-holomorphic Riemannian metrics and links them to para-Kähler Norden metrics, including natural Einstein metrics on semi-simple para-complex Lie groups.

Findings

Para-holomorphic Riemannian metrics correspond to para-Kähler Norden metrics.

The Einstein condition is characterized within this framework.

Semi-simple para-complex Lie groups naturally admit Einstein metrics.

Abstract

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