no. 7
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
Xu, Na1 ; Tan, Ju1
1 School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, People's Republic of China
Comptes Rendus. Mathématique, Tome 357 (2019) no. 7, pp. 624-628

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.07.003

Xu, Na 1 ; Tan, Ju 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 = {Xu, Na and Tan, Ju},
     title = {On left-invariant {Einstein} metrics that are not geodesic orbit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {624--628},
     year = {2019},
     publisher = {Elsevier},
     volume = {357},
     number = {7},
     doi = {10.1016/j.crma.2019.07.003},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2019.07.003/}
}
TY  - JOUR
AU  - Xu, Na
AU  - Tan, Ju
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
UR  - https://www.numdam.org/articles/10.1016/j.crma.2019.07.003/
DO  - 10.1016/j.crma.2019.07.003
LA  - en
ID  - CRMATH_2019__357_7_624_0
ER  - 
%0 Journal Article
%A Xu, Na
%A Tan, Ju
%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
%U https://www.numdam.org/articles/10.1016/j.crma.2019.07.003/
%R 10.1016/j.crma.2019.07.003
%G en
%F CRMATH_2019__357_7_624_0
Xu, Na; Tan, Ju. On left-invariant Einstein metrics that are not geodesic orbit. Comptes Rendus. Mathématique, Tome 357 (2019) no. 7, pp. 624-628. doi: 10.1016/j.crma.2019.07.003

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

[2] Chen, H.; Chen, Z.; Deng, S. 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] Chen, Z.; Liang, K. Non-naturally reductive Einstein metrics on the compact simple Lie group F4, Ann. Glob. Anal. Geom., Volume 46 (2014), pp. 103-115

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

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

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

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

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

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

Cité par Sources :