Article
| Title: | New Einstein metrics on ${\rm Sp}(n)$ which are non-naturally reductive (English) |
| Author: | Zhang, Shaoxiang |
| Author: | Chen, Huibin |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 2 |
| Year: | 2022 |
| Pages: | 349-363 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | 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. (English) |
| Keyword: | Einstein metric |
| Keyword: | non-naturally reductive metric |
| Keyword: | compact Lie group |
| Keyword: | symplectic group |
| MSC: | 53C25 |
| MSC: | 53C30 |
| MSC: | 65H10 |
| idZBL: | Zbl 07547208 |
| idMR: | MR4412763 |
| DOI: | 10.21136/CMJ.2021.0491-20 |
| . | |
| Date available: | 2022-04-21T18:59:13Z |
| Last updated: | 2022-09-08 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150405 |
| . | |
| Reference: | [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. Zbl 1183.53037, MR 2588419, 10.4153/CJM-2009-056-2 |
| Reference: | [2] Arvanitoyeorgos, A., Mori, K., Sakane, Y.: Einstein metrics on compact Lie groups which are not naturally reductive.Geom. Dedicate 160 (2012), 261-285. Zbl 1253.53043, MR 2970054, 10.1007/s10711-011-9681-1 |
| Reference: | [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. Zbl 1333.53053, MR 3494871, 10.1142/9789814719780_0001 |
| Reference: | [4] Besse, A. L.: Einstein Manifolds.Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer, Berlin (1987). Zbl 0613.53001, MR 0867684, 10.1007/978-3-540-74311-8 |
| Reference: | [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. Zbl 1386.53053, MR 3754514, 10.1016/j.geomphys.2017.09.011 |
| Reference: | [6] Chen, H., Chen, Z., Deng, S.: Non-naturally reductive Einstein metrics on SO$(n)$.Manuscr. Math. 156 (2018), 127-136. Zbl 1390.53040, MR 3783569, 10.1007/s00229-017-0954-3 |
| Reference: | [7] Chen, Z., Chen, H.: Non-naturally reductive Einstein metrics on Sp$(n)$.Front. Math. China 15 (2020), 47-55. Zbl 1447.53045, MR 4074344, 10.1007/s11464-020-0818-0 |
| Reference: | [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. Zbl 1301.53042, MR 3239276, 10.1007/s10455-014-9413-5 |
| Reference: | [9] Chrysikos, I., Sakane, Y.: Non-naturally reductive Einstein metrics on exceptional Lie groups.J. Geom. Phys. 116 (2017), 152-186. Zbl 1390.53041, MR 3623653, 10.1016/j.geomphys.2017.01.030 |
| Reference: | [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. Zbl 0404.53044, MR 0519928, 10.1090/memo/0215 |
| Reference: | [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. |
| Reference: | [12] Park, J.-S., Sakane, Y.: Invariant Einstein metrics on certain homogeneous spaces.Tokyo J. Math. 20 (1997), 51-61. Zbl 0884.53039, MR 1451858, 10.3836/tjm/1270042398 |
| Reference: | [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. Zbl 1003.53037, MR 1798614, 10.4310/SDG.2001.v6.n1.a11 |
| Reference: | [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. Zbl 1262.53044, MR 3076055 |
| Reference: | [15] Wang, M., Ziller, W.: Existence and non-existence of homogeneous Einstein metrics.Invent. Math. 84 (1986), 177-194. Zbl 0596.53040, MR 0830044, 10.1007/BF01388738 |
| Reference: | [16] Yan, Z., Deng, S.: Einstein metrics on compact simple Lie groups attached to standard triples.Trans. Am. Math. Soc. 369 (2017), 8587-8605. Zbl 1433.53079, MR 3710636, 10.1090/tran/7025 |
| Reference: | [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. MR 4245438, 10.1007/s10473-021-0315-x |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
