Einstein solvmanifolds have maximal symmetry

TL;DR

This paper proves that all known negatively curved homogeneous Einstein metrics on solvable Lie groups are maximally symmetric, advancing understanding of their geometric structure and symmetry properties.

Contribution

It introduces the concept of maximal symmetry for left-invariant metrics and proves that all negatively curved Einstein metrics on solvable Lie groups are maximally symmetric.

Findings

Negatively curved Einstein metrics are maximally symmetric.

Supports the Alekseevskii Conjecture.

Addresses existence of maximally symmetric metrics.

Abstract

All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riemannian metrics more generally.