Non-trivial $m$-quasi-Einstein metrics on simple Lie groups

TL;DR

This paper investigates $m$-quasi-Einstein metrics on simple Lie groups, establishing existence, invariance properties, and extending results to Lorentzian and non-compact cases, revealing many such metrics except for specific groups.

Contribution

It proves invariance of the vector field in compact cases, classifies existence of non-trivial $m$-quasi-Einstein metrics on simple Lie groups, and extends findings to Lorentzian and non-compact settings.

Findings

Compact simple Lie groups admit non-trivial $m$-quasi-Einstein metrics, except $SU(3)$, $E_8$, and $G_2$.

Most simple Lie groups have infinitely many such metrics.

Existence of non-trivial $m$-quasi-Einstein Lorentzian metrics on all compact simple Lie groups.

Abstract

We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that $X$ is a left-invariant Killing vector field if the metric on a compact simple Lie group is $m$-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial $m$-quasi-Einstein metrics except $SU(3)$, $E_8$ and $G_2$, and most of them admit infinitely many metrics. Naturally, the study on $m$-quasi-Einstein metrics can be extended to pseudo-Riemannian case. And we prove that every compact simple Lie group admits non-trivial $m$-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial $m$-quasi-Einstein Lorentzian metrics.