New Einstein metrics on ${\rm Sp}(n)$ which are non-naturally reductive

Zhang, Shaoxiang ; Chen, Huibin

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 349-363

Abstract (VO)

We prove that there are at least two new non-naturally reductive ${\rm Ad}({\rm Sp}(l)\times {\rm Sp}(k)\times {\rm Sp}(k)\times {\rm Sp}(k))$ invariant Einstein metrics on ${\rm Sp} (l+3k)$ $(k l)$. It implies that every compact simple Lie group ${\rm Sp} (n)$ for $n= l+3k>4$ admits at least $2[\tfrac 14 (n-1)]$ non-naturally reductive ${\rm Ad}({\rm Sp}(l)\times {\rm Sp}(k)\times {\rm Sp}(k)\times {\rm Sp}(k))$ invariant Einstein metrics.

Détail

DOI : 10.21136/CMJ.2021.0491-20

Classification : 53C25, 53C30, 65H10
Keywords: Einstein metric; non-naturally reductive metric; compact Lie group; symplectic group

Bibliographie

[1] Arvanitoyeorgos, A., Dzhepko, V. V., Nikonorov, Y. G.: Invariant Einstein metrics on some homogeneous spaces of classical Lie groups. Can. J. Math. 61 (2009), 1201-1213. | DOI | MR | JFM

[2] Arvanitoyeorgos, A., Mori, K., Sakane, Y.: Einstein metrics on compact Lie groups which are not naturally reductive. Geom. Dedicate 160 (2012), 261-285. | DOI | MR | JFM

[3] Arvanitoyeorgos, A., Sakane, Y., Statha, M.: Einstein metrics on the symplectic group which are not naturally reductive. Current Developments in Differential Geometry and its Related Fields World Scientific, Hackensack (2016), 1-22. | DOI | MR | JFM

[4] Besse, A. L.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer, Berlin (1987). | DOI | MR | JFM

[5] Chen, H., Chen, Z., Deng, S.: New non-naturally reductive Einstein metrics on exceptional simple Lie groups. J. Geom. Phys. 124 (2018), 268-285. | DOI | MR | JFM

[6] Chen, H., Chen, Z., Deng, S.: Non-naturally reductive Einstein metrics on SO$(n)$. Manuscr. Math. 156 (2018), 127-136. | DOI | MR | JFM

[7] Chen, Z., Chen, H.: Non-naturally reductive Einstein metrics on Sp$(n)$. Front. Math. China 15 (2020), 47-55. | DOI | MR | JFM

[8] Chen, Z., Liang, K.: Non-naturally reductive Einstein metrics on the compact simple Lie group $F_4$. Ann. Global Anal. Geom. 46 (2014), 103-115. | DOI | MR | JFM

[9] Chrysikos, I., Sakane, Y.: Non-naturally reductive Einstein metrics on exceptional Lie groups. J. Geom. Phys. 116 (2017), 152-186. | DOI | MR | JFM

[10] D'Atri, J. E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Am. Math. Soc. 215 (1979), 72 pages. | DOI | MR | JFM

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

[12] Park, J.-S., Sakane, Y.: Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20 (1997), 51-61. | DOI | MR | JFM

[13] Wang, M.: Einstein metrics from symmetry and bundle constructions. Surveys in Differential Geometry: Essays on Einstein Manifolds. Surv. Differ. Geom. VI International Press, Boston (1999), 287-325. | DOI | MR | JFM

[14] Wang, M.: Einstein metrics from symmetry and bundle constructions: A sequel. Differential Geometry: Under the Influence of S.-S. Chern Advanced Lectures in Mathematics 22. Higher Education Press/International Press, Somerville (2012), 253-309. | MR | JFM

[15] Wang, M., Ziller, W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84 (1986), 177-194. | DOI | MR | JFM

[16] Yan, Z., Deng, S.: Einstein metrics on compact simple Lie groups attached to standard triples. Trans. Am. Math. Soc. 369 (2017), 8587-8605. | DOI | MR | JFM

[17] Zhang, S., Chen, H., Deng, S.: New non-naturally reductive Einstein metrics on Sp$(n)$. Acta Math. Sci., Ser. B, Engl. Ed. 41 (2021), 887-898. | DOI | MR