Left-invariant Metrics on a Class of Solvable Lie Groups

Abstract

Left-invariant Riemannian metrics on Lie groups \(G_{{{\mathbb R}{\rm H}}^{n-1}} \times {\mathbb R}\) and \(G_{{{\mathbb R}{\rm H}}^{2}} \times {{\mathbb R}^{n-2}}\) (n ≥ 3) have been classified by Hiroshi, Takahara and Tamaru. It is easily seen that \(G_{{{\mathbb R}{\rm H}}^{n-1}} \times {\mathbb R}\) and \(G_{{{\mathbb R}{\rm H}}^{2}} \times {{\mathbb R}^{n-2}}\) (n ≥ 3) have the same automorphism group, which is denoted by L_n. In this paper, we first classify n-dimensional simply connected Lie groups with automorphism group L_n, then we classify the left-invariant Riemannian metrics on such Lie groups. As an application, we get the m-quasi Einstein metrics.