Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

Chen, Huibin; Chen, Zhiqi; Deng, Shaoqiang

doi:10.1016/j.crma.2017.11.018

Geometry

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit
[Groupes de Lie simples compacts admettant des métriques d'Einstein invariantes à gauche, dont une géodésique n'est pas une orbite]

Présenté par : Claire Voisin

Chen, Huibin ¹ ; Chen, Zhiqi ¹ ; Deng, Shaoqiang ¹

¹ School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China

Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 81-84.

Résumé
Abstract

Dans cet article, nous démontrons que les groupes simples compacts $S U (n)$ pour $n \geq 6$ , $S O (n)$ pour $n \geq 7$ , $S p (n)$ pour $n \geq 3$ , $E_{6}$ , $E_{7}$ , $E_{8}$ et $F_{4}$ admettent des métriques d'Einstein invariantes à gauche, dont une géodésique maximale n'est pas une orbite d'un sous-groupe à un paramètre du groupe des isométries complet. Ceci fournit une réponse positive à un problème récemment posé par Nikonorov.

In this article, we prove that the compact simple Lie groups $S U (n)$ for $n \geq 6$ , $S O (n)$ for $n \geq 7$ , $S p (n)$ for $n \geq 3$ , $E_{6}, E_{7}, E_{8}$ , and $F_{4}$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.

Reçu le : 2017-10-18
Accepté le : 2017-11-30
Publié le : 2017-12-06

DOI : 10.1016/j.crma.2017.11.018

Affiliations des auteurs :

Chen, Huibin ¹ ; Chen, Zhiqi ¹ ; Deng, Shaoqiang ¹

¹ School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China

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     author = {Chen, Huibin and Chen, Zhiqi and Deng, Shaoqiang},
     title = {Compact simple {Lie} groups admitting left-invariant {Einstein} metrics that are not geodesic orbit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {81--84},
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     doi = {10.1016/j.crma.2017.11.018},
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Chen, Huibin; Chen, Zhiqi; Deng, Shaoqiang. Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit. Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 81-84. doi : 10.1016/j.crma.2017.11.018. http://www.numdam.org/articles/10.1016/j.crma.2017.11.018/

Bibliographie
Cité par

[1] Arvanitoyeorgos, A.; Sakane, Y.; Statha, M. Einstein metrics on the symmetric group which are not naturally reductive, Velico Tarnovo, Bulgaria 2014, World Scientific (2015), pp. 1-22

[2] Arvanitoyeorgos, A.; Sakane, Y.; Statha, M. New Einstein metrics on the Lie group $S O (n)$ which are not naturally reductive, Geom. Imaging Comput., Volume 2 (2015) no. 2, pp. 77-108

[3] Chen, H.; Chen, Z.; Deng, S. New non-naturally reductive Einstein metrics on exceptional simple Lie groups, 2016 (Preprint) | arXiv

[4] Chen, Z.; Liang, K. Non-naturally reductive Einstein metrics on the compact simple Lie group $F_{4}$ , Ann. Glob. Anal. Geom., Volume 46 (2014), pp. 103-115

[5] Chen, Z.; Kang, Y.; Liang, K. Invariant Einstein metrics on three-locally-symmetric spaces, Commun. Anal. Geom., Volume 24 (2016) no. 4, pp. 769-792

[6] Chrysikos, I.; Sakane, Y. Non-naturally reductive Einstein metrics on exceptional Lie groups, J. Geom. Phys., Volume 116 (2017), pp. 152-186

[7] Kowalski, O.; Vanhecke, L. Riemannian manifolds with homogeneous geodesics, Boll. Unione Mat. Ital., B (7), Volume 5 (1991) no. 1, pp. 189-246

[8] Mori, K. Left Invariant Einstein Metrics on $S U (n)$ That Are Not Naturally Reductive, Osaka University, 1994 Master Thesis (in Japanese) English Translation: Osaka University RPM 96010 (preprint series), 1996

[9] Nikonorov, Y.G. Classification of generalized Wallach spaces, Geom. Dedic., Volume 181 (2016) no. 1, pp. 193-212

[10] Nikonorov, Y.G. On left-invariant Einstein Riemannian metrics that are not geodesic orbit, Transform. Groups (2017) (in press) | arXiv

[11] Zhang, L.; Chen, Z.; Deng, S. New Einstein metrics on $E_{7}$ , Differ. Geom. Appl., Volume 51 (2017), pp. 189-202

Cité par Sources :

^☆ Supported by NSFC (No. 11671212, 51535008) of China.