Geodesic
Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups
Archivum mathematicum, Tome 50 (2014) no. 3, pp. 171-192 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $N_{3,2}$.
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 $N_{3,2}$.
DOI : 10.5817/AM2014-3-171
Classification : 22E25, 53C25, 53C50
Keywords: 2-step nilpotent Lie groups; free nilpotent groups; left-invariant Lorentzian metrics; Ricci-flatness 
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     title = {Ricci-flat left-invariant {Lorentzian} metrics on 2-step nilpotent {Lie} groups},
     journal = {Archivum mathematicum},
     pages = {171--192},
     year = {2014},
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Guediri, Mohammed; Bin-Asfour, Mona. Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. Archivum mathematicum, Tome 50 (2014) no. 3, pp. 171-192. doi: 10.5817/AM2014-3-171

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