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

Mohammed Guediri; Mona Bin-Asfour

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 3, page 171-192
  • ISSN: 0044-8753

Abstract

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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 , is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group N , then the restriction of , 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 2 n + 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 .

How to cite

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Guediri, Mohammed, and Bin-Asfour, Mona. "Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups." Archivum Mathematicum 050.3 (2014): 171-192. <http://eudml.org/doc/261970>.

@article{Guediri2014,
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 $\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 $N_\{3,2\}$.},
author = {Guediri, Mohammed, Bin-Asfour, Mona},
journal = {Archivum Mathematicum},
keywords = {2-step nilpotent Lie groups; free nilpotent groups; left-invariant Lorentzian metrics; Ricci-flatness; 2-step nilpotent Lie groups; free nilpotent groups; left-invariant Lorentzian metrics; Ricci-flatness},
language = {eng},
number = {3},
pages = {171-192},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups},
url = {http://eudml.org/doc/261970},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Guediri, Mohammed
AU - Bin-Asfour, Mona
TI - Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 3
SP - 171
EP - 192
AB - 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 $N_{3,2}$.
LA - eng
KW - 2-step nilpotent Lie groups; free nilpotent groups; left-invariant Lorentzian metrics; Ricci-flatness; 2-step nilpotent Lie groups; free nilpotent groups; left-invariant Lorentzian metrics; Ricci-flatness
UR - http://eudml.org/doc/261970
ER -

References

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