Classification of left-invariant Einstein metrics on $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ that are bi-invariant under a one-parameter subgroup

TL;DR

This paper classifies all specific Einstein metrics on the product of two SL(2,R) groups that are invariant under a one-parameter subgroup, identifying only two such metrics: the Killing form and a nearly pseudo-Kähler metric.

Contribution

It provides a complete classification of left-invariant pseudo-Riemannian Einstein metrics on SL(2,R)×SL(2,R) with bi-invariance under a one-parameter subgroup, revealing only two such metrics.

Findings

Identified exactly two Einstein metrics under the specified conditions.

The metrics are the Killing form and a nearly pseudo-Kähler metric.

No other such metrics exist up to homothety.

Abstract

We classify all left-invariant pseudo-Riemannian Einstein metrics on $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ that are bi-invariant under a one-parameter subgroup. We find that there are precisely two such metrics up to homothety, the Killing form and a nearly pseudo-K\"ahler metric.