Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

Abstract

We call a metric m-quasi-Einstein if \({Ric_X^m}\) (a modification of the m-Bakry–Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contain Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field X such that the left-invariant Riemannian metric on a quadratic Lie group is m-quasi-Einstein is a Killing vector field. Then we construct infinitely many non-trivial m-quasi-Einstein metrics on solvable quadratic Lie groups G(n) for m finite.