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

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

2009-10-14

arxiv.org/abs/0910.2563v2

Mathematics

Differential Geometry

Title modified, some explicites examples are given and Section 6 deleted

Scientific paper

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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