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**Einstein solvmanifolds have maximal symmetry.**
*(English)*
Zbl 1415.53037

The main result of the paper under review is the observation that left-invariant Einstein metrics with negative Ricci curvature on solvable Lie groups satisfy one strong geometric property. It should be recalled that all known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups.

In order to state the main result, the authors introduce the following definition (Definition 0.2): A left-invariant Riemannian metric \(g\) on a Lie group \(G\) is said to be maximally symmetric if for any other left-invariant Riemannian metric \(h\) on \(G\) we have \(\operatorname{Isom}(G, h) \subset \operatorname{Isom}(G, \psi^* g)\) for some \(\psi \in \operatorname{Aut}(G)\). There are examples of Lie groups that admit (resp. do not admit) maximally symmetric left-invariant Riemannian metrics. Therefore, there are at least two natural questions: (1) Which Lie groups admit maximally symmetric left-invariant metrics? (2) Are left-invariant metrics that are geometrically distinguished, e.g., those with special curvature properties, maximally symmetric? The discussion of question (1) could be found in Section 1 of the paper, while most of the paper is devoted to the study of question (2).

The main result (Theorem 0.1) of the paper under review is the following: If a solvable Lie group \(S\) admits a left-invariant Einstein metric \(g\) of negative Ricci curvature, then \(g\) is maximally symmetric.

Hence, this result implies an important (partial) affirmative answer to the above question (2). It is also closely related to the Alekseevskii Conjecture (given a homogeneous Einstein space \(G/K\) with negative scalar curvature, \(K\) must be a maximal compact subgroup of \(G\)) and to the question of stability of Einstein metrics under the Ricci flow. The proof of the above theorem is very difficult from the technical point of view. For this goal, the authors apply deep results concerning the structure of Einstein solvmanifolds, isometry groups of solvmanifolds, nilsolitons, and geometric invariant theory. Sections 2–5 are devoted to a comprehensive proof of the main result. In the final Section 6, the authors discuss Einstein and Ricci soliton extensions for solvable Lie groups and clarify some technical details of the study.

In order to state the main result, the authors introduce the following definition (Definition 0.2): A left-invariant Riemannian metric \(g\) on a Lie group \(G\) is said to be maximally symmetric if for any other left-invariant Riemannian metric \(h\) on \(G\) we have \(\operatorname{Isom}(G, h) \subset \operatorname{Isom}(G, \psi^* g)\) for some \(\psi \in \operatorname{Aut}(G)\). There are examples of Lie groups that admit (resp. do not admit) maximally symmetric left-invariant Riemannian metrics. Therefore, there are at least two natural questions: (1) Which Lie groups admit maximally symmetric left-invariant metrics? (2) Are left-invariant metrics that are geometrically distinguished, e.g., those with special curvature properties, maximally symmetric? The discussion of question (1) could be found in Section 1 of the paper, while most of the paper is devoted to the study of question (2).

The main result (Theorem 0.1) of the paper under review is the following: If a solvable Lie group \(S\) admits a left-invariant Einstein metric \(g\) of negative Ricci curvature, then \(g\) is maximally symmetric.

Hence, this result implies an important (partial) affirmative answer to the above question (2). It is also closely related to the Alekseevskii Conjecture (given a homogeneous Einstein space \(G/K\) with negative scalar curvature, \(K\) must be a maximal compact subgroup of \(G\)) and to the question of stability of Einstein metrics under the Ricci flow. The proof of the above theorem is very difficult from the technical point of view. For this goal, the authors apply deep results concerning the structure of Einstein solvmanifolds, isometry groups of solvmanifolds, nilsolitons, and geometric invariant theory. Sections 2–5 are devoted to a comprehensive proof of the main result. In the final Section 6, the authors discuss Einstein and Ricci soliton extensions for solvable Lie groups and clarify some technical details of the study.

Reviewer: Yurii G. Nikonorov (Volgodonsk)

### MSC:

53C30 | Differential geometry of homogeneous manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

22E25 | Nilpotent and solvable Lie groups |