Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

Abstract

There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group $H_{2n+1}$, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra ${\mathfrak{h}}_{2n+1}$, denoted by $\mu$, $\nu$, and $\varphi$. We show that, in a specific case, $\mu$ is an algebraic Ricci soliton that is shrinking. Moreover, $\nu$ is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra ${\mathfrak{h}}_3$ and it is shrinking. Finally, we show that $\varphi$ is a steady algebraic Ricci soliton on ${\mathfrak{h}}_{2n+1}$ for $n>1$. However, for $n=1$, $\varphi$ is flat.