The scalar curvature functional and homogeneous Einsteinian metrics on Lie groups
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 \(G \rightarrow G^n\).
scientific article; zbMATH DE number 1225847
scientific article