Abstract
AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.
Highlights
This is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed
A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group
The construction of Riemannian Einstein solvmanifolds is reduced to the study of the Ricci operator on a nilpotent Lie algebra, as they are characterized by the so-called nilsoliton equation ([22]), involving the Ricci operator of the metric restricted to the nilradical
Summary
A considerable amount of research has been devoted to the classi cation of low-dimensional nilsolitons (see [2, 12, 18, 23, 26, 27, 32]; more references can be found in [21]); most of these results employ, directly or indirectly, the notion of a nice basis. In this paper we determine conditions on a nice Lie algebra that are both necessary and su cient for the existence of an Einstein metric of diagonal or σ-diagonal type (Theorems 2.2 and 2.7). These conditions are still polynomial, but they involve a lower number of parameters and equations than (1). We apply this criterion to the case λ = , obtaining a classi cation of diagonal and σ-diagonal Ricci- at metrics on nice nilpotent Lie algebras of dimension ≤. The nal section is dedicated to the Ricci- at case; it contains a classi cation of diagonal and σ-diagonal Ricci- at metrics on nice nilpotent Lie algebras of dimension ≤ , as well as some remarks on the -step case and examples related to parahermitian geometry
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