Conformal vector fields on Lie groups

TL;DR

This paper studies conformal vector fields on Lie groups with pseudo-Riemannian metrics, showing that on unimodular groups they are Killing, and exploring conditions and examples for non-unimodular groups.

Contribution

It proves that conformal vector fields on unimodular Lie groups are Killing and provides conditions and examples for non-unimodular groups with non-Killing conformal fields.

Findings

Conformal vector fields on unimodular Lie groups are Killing.

Necessary conditions for non-unimodular groups admitting non-Killing conformal fields.

Examples of non-Killing conformal vector fields and Yamabe solitons on non-unimodular Lorentzian Lie groups.

Abstract

In this paper, we investigated the behavior of left-invariant conformal vector fields on Lie groups with left-invariant pseudo-Riemannian metrics. First of all, we prove that conformal vector fields on pseudo-Riemannian unimodular Lie groups are Killing. Then we obtain a necessary condition for a pseudo-Rimennian non-unimodular Lie group admitting a non-Killing conformal vector field. Finally, we give examples of non-Killing conformal vector fields and Yamabe solitons on non-unimodular Lorentzian Lie groups based on the above study.