Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

TL;DR

This paper classifies Lorentz Ricci solitons on 4-dimensional non-Abelian nilpotent Lie groups, identifying specific metrics that satisfy the Ricci soliton equation and their shrinking or expanding nature.

Contribution

It provides a detailed classification of Lorentz Ricci solitons on certain 4D nilpotent Lie groups, highlighting which metrics satisfy the Ricci soliton condition.

Findings

Identifies specific metrics on $H_3 imes R$ that are Ricci solitons.

Determines which metrics on $G_4$ satisfy Ricci soliton equations.

Classifies shrinking and expanding Ricci solitons among these metrics.

Abstract

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3 \times {\Bbb R}$, ~ $g_{\lambda}^{+}$ is a shrinking while $g_{\lambda}^{+}$ and $g_{\mu}$ are expanding and also $g_0^1, g_0^2, g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^\lambda, g_2^\lambda$ have Lorentz Ricci solitons and $g_2^\lambda$ is a shrinking.