Every nilpotent Lie group of dimension at most 6, every nice nilpotent Lie group of dimension at most 7, and every two-step nilpotent Lie group associated to a graph admits an indefinite Ricci-flat metric, almost always nonflat.
Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups
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abstract
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a Ricci flat left invariant pseudo-Riemannian metric of signature $(q,2k+1-q)$ and we give explicite examples of such metrics.
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math.DG 1years
2019 1verdicts
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Diagram involutions and homogeneous Ricci-flat metrics
Every nilpotent Lie group of dimension at most 6, every nice nilpotent Lie group of dimension at most 7, and every two-step nilpotent Lie group associated to a graph admits an indefinite Ricci-flat metric, almost always nonflat.