Abstract
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 <,> i is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group N, then the restriction of h, i 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 H2n+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,N-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,N-2.