Home Mathematics Left invariant Ricci flat metrics on Lie groups
Article
Licensed
Unlicensed Requires Authentication

Left invariant Ricci flat metrics on Lie groups

  • and EMAIL logo
Published/Copyright: April 27, 2023

Abstract

In this paper, we apply the double extension process to study left invariant Ricci flat metrics on solvable and non-solvable Lie groups. An inductive method to produce new Ricci flat metrics from the old ones is established. As applications, we prove the following two results: (i) Every nilpotent Lie group with dim C ( G ) 1 2 ( dim G - 1 ) admits a left invariant Ricci flat metric. (ii) Given a Lie group G, there exists a nilpotent Lie group N with nilpotent index at most 2 such that G × N admits a left invariant Ricci flat metric. We also construct infinitely many new explicit examples of left invariant Ricci flat metrics on nilpotent Lie groups.

MSC 2020: 22E15; 22E25; 53C25

Communicated by Karin Melnick


Award Identifier / Grant number: 12131012

Award Identifier / Grant number: 12071228

Funding statement: Zaili Yan is supported by the Fundamental Research Funds for the Provincial Universities of Zhejiang and K. C. Wong Magna Fund in Ningbo University. Shaoqiang Deng is supported by NSFC (nos. 12131012, 12071228), and the Fundamental Research Funds for the Central Universities.

Acknowledgements

We are deeply grateful to the reviewers of this paper for very careful reading and useful suggestions.

References

[1] M. Ait Ben Haddou, M. Boucetta and H. Lebzioui, Left-invariant Lorentzian flat metrics on Lie groups, J. Lie Theory 22 (2012), no. 1, 269–289. Search in Google Scholar

[2] D. V. Alekseevskiĭ and B. N. Kimelfeld, Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funct. Anal. Appl. 9 (1975), 97–102. 10.1007/BF01075445Search in Google Scholar

[3] A. Aubert and A. Medina, Groupes de Lie pseudo-riemanniens plats, Tohoku Math. J. (2) 55 (2003), no. 4, 487–506. 10.2748/tmj/1113247126Search in Google Scholar

[4] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. (N. S.) 66 (1997), no. 2, 151–201. Search in Google Scholar

[5] M. Boucetta, Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups, preprint (2010), https://arxiv.org/abs/0910.2563v2. Search in Google Scholar

[6] M. Boucetta and H. Lebzioui, Flat nonunimodular Lorentzian Lie algebras, Comm. Algebra 44 (2016), no. 10, 4185–4195. 10.1080/00927872.2015.1087016Search in Google Scholar

[7] M. Boucetta and H. Lebzioui, On flat pseudo-Euclidean nilpotent Lie algebras, J. Algebra 537 (2019), 459–477. 10.1016/j.jalgebra.2019.07.018Search in Google Scholar

[8] M. Boucetta and O. Tibssirte, On Einstein Lorentzian nilpotent Lie groups, J. Pure Appl. Algebra 224 (2020), no. 12, Article ID 106443. 10.1016/j.jpaa.2020.106443Search in Google Scholar

[9] S. Chen and K. Liang, Left-invariant pseudo-Einstein metrics on Lie groups, J. Nonlinear Math. Phys. 19 (2012), no. 2, Article ID 1250015. 10.1142/S1402925112500155Search in Google Scholar

[10] D. Conti, V. del Barco and F. A. Rossi, Diagram involutions and homogeneous Ricci-flat metrics, Manuscripta Math. 165 (2021), no. 3–4, 381–413. 10.1007/s00229-020-01225-ySearch in Google Scholar

[11] D. Conti and F. A. Rossi, Construction of nice nilpotent Lie groups, J. Algebra 525 (2019), 311–340. 10.1016/j.jalgebra.2019.01.020Search in Google Scholar

[12] D. Conti and F. A. Rossi, Einstein nilpotent Lie groups, J. Pure Appl. Algebra 223 (2019), no. 3, 976–997. 10.1016/j.jpaa.2018.05.010Search in Google Scholar

[13] D. Conti and F. A. Rossi, Ricci-flat and Einstein pseudoriemannian nilmanifolds, Complex Manifolds 6 (2019), no. 1, 170–193. 10.1515/coma-2019-0010Search in Google Scholar

[14] D. Conti and F. A. Rossi, Indefinite Einstein metrics on nice Lie groups, Forum Math. 32 (2020), no. 6, 1599–1619. 10.1515/forum-2020-0049Search in Google Scholar

[15] G. Favre and L. J. Santharoubane, Symmetric, invariant, nondegenerate bilinear form on a Lie algebra, J. Algebra 105 (1987), no. 2, 451–464. 10.1016/0021-8693(87)90209-2Search in Google Scholar

[16] M. Guediri and M. Bin-Asfour, Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Liegroups, Arch. Math. (Brno) 50 (2014), no. 3, 171–192. 10.5817/AM2014-3-171Search in Google Scholar

[17] I. Kath, Nilpotent metric Lie algebras of small dimension, J. Lie Theory 17 (2007), no. 1, 41–61. Search in Google Scholar

[18] H. Lebzioui, Flat left-invariant pseudo-Riemannian metrics on unimodular Lie groups, Proc. Amer. Math. Soc. 148 (2020), no. 4, 1723–1730. 10.1090/proc/14808Search in Google Scholar

[19] A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant, Ann. Sc. Éc. Norm. Supér. (4) 18 (1985), no. 3, 553–561. 10.24033/asens.1496Search in Google Scholar

[20] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), no. 3, 293–329. 10.1016/S0001-8708(76)80002-3Search in Google Scholar

[21] K. Nomizu, Left-invariant Lorentz metrics on Lie groups, Osaka Math. J. 16 (1979), no. 1, 143–150. Search in Google Scholar

[22] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure Appl. Math. 103, Academic Press, New York, 1983. Search in Google Scholar

[23] G. P. Ovando, Lie algebras with ad-invariant metrics: A survey-guide, Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), no. 1, 243–268. Search in Google Scholar

[24] Y. Xiang and Z. Yan, Existence of left invariant Ricci flat metrics on nilpotent Lie groups, Arch. Math. (Basel) 117 (2021), no. 5, 569–578. 10.1007/s00013-021-01645-6Search in Google Scholar

[25] Z. Yan, Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces, J. Geom. 111 (2020), no. 1, Paper No. 4. 10.1007/s00022-019-0518-7Search in Google Scholar

[26] Z. Yan and S. Deng, Double extensions on Riemannian Ricci nilsolitons, J. Geom. Anal. 31 (2021), no. 10, 9996–10023. 10.1007/s12220-021-00636-xSearch in Google Scholar

Received: 2022-04-03
Revised: 2023-01-15
Published Online: 2023-04-27
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0102/html
Scroll to top button