Left invariant pseudo-Riemannian metrics on solvable Lie groups

Abstract

In this paper, we consider a special class of solvable Lie groups such that for any x,y in its Lie algebra, [x,y] is a linear combination of x andy. For the convenience, we call such a Lie group a LCS Lie group. We investigate non-trivial m-quasi-Einstein metrics on pseudo-Riemannian LCS Lie group. We proved that although there exists only trivial Ricci soliton on pseudo-Riemannian LCS Lie group, any left invariant pseudo-Riemannian metric on LCS Lie group is non-trivial m-quasi-Einstein metric. Moreover, non-trivial m-quasi-Einstein metric is shrinking, expanding, or steady if LCS Lie group has positive, negative, or zero constant sectional curvature respectively. In particular, any left invariant Riemannian or Lorentzian metric on LCS Lie group is non-trivial m-quasi-Einstein metric.