On left-invariant Einstein metrics that are not geodesic orbit

Na Xu; Ju Tan

doi:10.1016/j.crma.2019.07.003

Differential geometry/Lie algebras

On left-invariant Einstein metrics that are not geodesic orbit
[Sur les métriques d'Einstein invariantes à gauche, qui ne sont pas à orbites géodésiques]

Présenté par : the Editorial Board

Na Xu ¹ ; Ju Tan ¹

¹ School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, People's Republic of China

Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 624-628.

Résumé
Abstract

Dans cette Note, nous démontrons que les groupes de Lie simples, compacts, $SO (n)$ ( $n > 12$ ) admettent au moins deux métriques d'Einstein invariantes à gauche, dont des géodésiques maximales ne sont pas des orbites de sous-groupes à un paramètre du groupe d'isométries complet. Ceci répond par l'affirmative à une question récemment posée par Nikonorov.

In this article, we prove that compact simple Lie groups $SO (n)$ ( $n > 12$ ) admit at least two left-invariant Einstein metrics that are not geodesic orbit, which gives a positive answer to a problem recently posed by Nikonorov.

Reçu le : 2019-04-02
Accepté le : 2019-07-12
Publié le : 2019-07-01

DOI : 10.1016/j.crma.2019.07.003

Affiliations des auteurs :

Na Xu ¹ ; Ju Tan ¹

¹ School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, People's Republic of China

@article{CRMATH_2019__357_7_624_0,
     author = {Na Xu and Ju Tan},
     title = {On left-invariant {Einstein} metrics that are not geodesic orbit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {624--628},
     publisher = {Elsevier},
     volume = {357},
     number = {7},
     year = {2019},
     doi = {10.1016/j.crma.2019.07.003},
     language = {en},
}

TY  - JOUR
AU  - Na Xu
AU  - Ju Tan
TI  - On left-invariant Einstein metrics that are not geodesic orbit
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 624
EP  - 628
VL  - 357
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2019.07.003
LA  - en
ID  - CRMATH_2019__357_7_624_0
ER  -

%0 Journal Article
%A Na Xu
%A Ju Tan
%T On left-invariant Einstein metrics that are not geodesic orbit
%J Comptes Rendus. Mathématique
%D 2019
%P 624-628
%V 357
%N 7
%I Elsevier
%R 10.1016/j.crma.2019.07.003
%G en
%F CRMATH_2019__357_7_624_0

Na Xu; Ju Tan. On left-invariant Einstein metrics that are not geodesic orbit. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 624-628. doi : 10.1016/j.crma.2019.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.07.003/

Bibliographie
Cité par

[1] A. Arvanitoyeorgos; K. Mori; Y. Sakane Einstein metrics on compact Lie groups which are not naturally reductive, Geom. Dedic., Volume 160 (2012) no. 1, pp. 261-285

[2] H. Chen; Z. Chen; S. Deng Compact simple Lie groups admitting left invariant Einstein metrics that are not geodesic orbit, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2018), pp. 81-84

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

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

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

[6] Y.G. Nikonorov On left invariant Einstein Riemannian metrics that are not geodesic orbit, Transform. Groups, Volume 24 (2019) no. 2, pp. 511-530

[7] J. Ta; N. Xu Homogeneous Einstein–Randers metrics on symplectic groups, J. Math. Anal. Appl., Volume 472 (2019), pp. 1902-1913

[8] Z. Ya; S. Deng Einstein metrics on compact simple Lie groups attached to standard triples, Trans. Amer. Math. Soc., Volume 369 (2017), pp. 8587-8605

[9] B. Zhang; H. Chen; J. Tan New non-naturally reductive Einstein metrics on SO(n), Int. J. Math., Volume 29 (2018)

Nikolaos Panagiotis Souris Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups, Communications in Contemporary Mathematics, Volume 27 (2025) no. 1, p. 29 (Id/No 2350068) | DOI:10.1142/s0219199723500682 | Zbl:7939905
Wenyan Luo; Ju Tan; Na Xu New non-geodesic orbit Einstein metrics on $S p (n)$ , Manuscripta Mathematica, Volume 176 (2025) no. 1, p. 23 (Id/No 13) | DOI:10.1007/s00229-025-01614-1 | Zbl:7989704

Cité par 2 documents. Sources : zbMATH

Commentaires - Politique