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

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 $2\le q\le k$, $H_{2k+1}$ carries a Ricci flat left invariant pseudo-Riemannian metric of signature $(q,2k+1-q)$ and we give explicite examples of such metrics.

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

Cited by in corpus (3)

• Ricci-flat and Einstein pseudoriemannian nilmanifolds
• Naturally reductive pseudo-Riemannian Lie groups in low dimensions
• Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups