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 (Q2689728)

The long-standing Alekseevsky conjecture has been recently proved by \textit{C. Böhm} and \textit{R. Lafuente} [``Non-compact Einstein manifolds with symmetry'', Preprint, \url{arXiv:2107.04210}]. It ensures that every connected homogeneous Riemannian Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space. In particular, the non-compact Lie group \(\operatorname{SL}(2,\mathbb R)\times \operatorname{SL}(2,\mathbb R)\) does not admit any left-invariant Riemannian Einstein metric. In the article under review, the authors study left-invariant pseudo-Riemannian Einstein metrics on \(\operatorname{SL}(2,\mathbb R)\times \operatorname{SL}(2,\mathbb R)\). They proved that the only such metrics that are bi-invariant under a one-parameter subgroup are, up to homotheties, the one induced by the Killing form and a nearly pseudo-Kähler metric already known. As it is usual in several classifications of invariant Einstein metrics, the proof uses Gröbner bases and computer assisted calculations. This article seems to be a continuation of [\textit{F. Belgun} et al., J. Geom. Phys. 128, 128--139 (2018, Zbl 1403.53038)], by three of the authors, where the compact dual case \(\operatorname{SU}(2)\times \operatorname{SU}(2)\) was considered (in the Riemannian setting).