| Title:
|
Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups (English) |
| Author:
|
Guediri, Mohammed |
| Author:
|
Bin-Asfour, Mona |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
171-192 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $N_{3,2}$. (English) |
| Keyword:
|
2-step nilpotent Lie groups |
| Keyword:
|
free nilpotent groups |
| Keyword:
|
left-invariant Lorentzian metrics |
| Keyword:
|
Ricci-flatness |
| MSC:
|
22E25 |
| MSC:
|
53C25 |
| MSC:
|
53C50 |
| idZBL:
|
Zbl 06487005 |
| idMR:
|
MR3263659 |
| DOI:
|
10.5817/AM2014-3-171 |
| . |
| Date available:
|
2014-09-20T17:12:35Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143925 |
| . |
| Reference:
|
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