Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

summary:The purpose of this paper is to investigate Ricci-flatness of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. We first show that if $\left\langle \, ,\right\rangle $ is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group $N$, then the restriction of $\left\langle \, ,\right\rangle $ to the center of the Lie algebra of $N$ is degenerate. We then characterize the 2-step nilpotent Lie groups which can be endowed with a Ricci-flat left-invariant Lorentzian metric, and we deduce from this that a Heisenberg Lie group $H_{2n+1}$ can be endowed with Ricci-flat left-invariant Lorentzian metric if and only if $n=1$. We also show that the free 2-step nilpotent Lie group on $m$ generators $N_{m,2}$ admits a Ricci-flat left-invariant Lorentzian metric if and only if $m=2$ or $m=3$, and we determine all Ricci-flat left-invariant Lorentzian metrics on the free $2$-step nilpotent Lie group on $3$ generators $N_{3,2}$