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

Abstract

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3\times \mathbb{R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3\times \mathbb{R}$, ~ $g_{\lambda }^{+}$ 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.