The scalar curvature functional and homogeneous Einsteinian metrics on Lie groups
type: Item
instance of: scholarly article
title: The scalar curvature functional and homogeneous Einsteinian metrics on Lie groups
zbMATH Open document ID: 0905.53033
DOI: 10.1007/BF02673906
published in: Siberian Mathematical Journal
publication date: 1998-11-22T00:00:00Z
review text: Let \(H \subset K \subset G\) be a sequence of embeddings of Lie groups, with \(G\) an unimodular group, and let \(h \subset k \subset g\) be the corresponding sequence of Lie algebras of these groups. Denote by \(p\) an \(\text{ad }k\)-invariant complement to \(h\) in \(g\). The left-invariant metrics on \(G/H\) are in one-to-one correspondence with the \(\text{ad }h\)-invariant inner products on \(p\). For the case that \(G\) is a unimodular group and \(H\) is a unit group, \textit{G. R. Jensen} [Indiana Univ. Math. J. 20, 1125-1144 (1971; Zbl 0219.53044)] considered a set \(M\) of such inner products normalized by an additional condition and proved that the left-invariant Einstein metrics on \(G = G/H\) are in one-to-one correspondence with the critical points of the scalar curvature functional on \(M\). The author extends this result to the general case of \(\text{Ad }K\)-invariant Einstein metrics on \(G/H\). As an application, he describes the Einstein metrics on products of compact groups \(G^n = G \times \dots \times G\) invariant under the image of the diagonal embedding \(G \rightarrow G^n\).
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1225847
zbMATH Keywords: homogeneous space
zbMATH Keywords: Einstein metric
zbMATH Keywords: scalar curvature functional
author: Yurii G. Nikonorov
reviewed by: I. A. Taimanov
MaRDI profile type: Publication
Wikidata QID: Q115391909
cites work: Q4002725
cites work: Q5624353
cites work: Q3671396
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description: scientific article; zbMATH DE number 1225847