Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Einstein metrics on compact simple Lie groups attached to standard triples


Authors: Zaili Yan and Shaoqiang Deng
Journal: Trans. Amer. Math. Soc. 369 (2017), 8587-8605
MSC (2010): Primary 53C25, 53C35, 53C30
DOI: https://doi.org/10.1090/tran/7025
Published electronically: May 5, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study left invariant Einstein metrics on compact simple Lie groups. We find a method to construct left invariant non-naturally reductive Einstein metrics on compact simple Lie groups from a standard triple. This result, combined with the classification of standard homogeneous Einstein manifolds, leads to a large number of new Einstein metrics on compact simple Lie groups which are not naturally reductive. In particular, we show that on the compact simple Lie groups $ \mathrm {SO}(8)$ and $ \mathrm {SO}(10)$, there exist Einstein metrics which are not naturally reductive. A further interesting result of this paper is that on the simple Lie groups $ \mathrm {SO}(2n)$ and $ \mathrm {Sp}(2n)$ there exist a large number of left invariant non-naturally reductive Einstein metrics.


References [Enhancements On Off] (What's this?)

  • [1] Andreas Arvanitoyeorgos, Kunihiko Mori, and Yusuke Sakane, Einstein metrics on compact Lie groups which are not naturally reductive, Geom. Dedicata 160 (2012), 261-285. MR 2970054, https://doi.org/10.1007/s10711-011-9681-1
  • [2] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • [3] Christoph Böhm, Homogeneous Einstein metrics and simplicial complexes, J. Differential Geom. 67 (2004), no. 1, 79-165. MR 2153482
  • [4] Christoph Böhm and Megan M. Kerr, Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1455-1468. MR 2186982, https://doi.org/10.1090/S0002-9947-05-04096-1
  • [5] C. Böhm, M. Wang, and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (2004), no. 4, 681-733. MR 2084976, https://doi.org/10.1007/s00039-004-0471-x
  • [6] Zhiqi Chen and Ke Liang, Non-naturally reductive Einstein metrics on the compact simple Lie group $ F_4$, Ann. Global Anal. Geom. 46 (2014), no. 2, 103-115. MR 3239276, https://doi.org/10.1007/s10455-014-9413-5
  • [7] J. E. D'Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215, iii+72. MR 519928
  • [8] William Dickinson and Megan M. Kerr, The geometry of compact homogeneous spaces with two isotropy summands, Ann. Global Anal. Geom. 34 (2008), no. 4, 329-350. MR 2447903, https://doi.org/10.1007/s10455-008-9109-9
  • [9] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349-462; English transl., Translations Amer. Math. Soc., ser. 2, 6 (1957), 111-244 (Russian). MR 0047629
  • [10] G. W. Gibbons, H. Lü, and C. N. Pope, Einstein metrics on group manifolds and cosets, J. Geom. Phys. 61 (2011), no. 5, 947-960. MR 2776894, https://doi.org/10.1016/j.geomphys.2011.01.004
  • [11] Gary R. Jensen, Einstein metrics on principal fibre bundles, J. Differential Geometry 8 (1973), 599-614. MR 0353209
  • [12] Jens Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), no. 2, 279-352. MR 1632782, https://doi.org/10.1007/s002220050247
  • [13] Jorge Lauret, Einstein solvmanifolds are standard, Ann. of Math. (2) 172 (2010), no. 3, 1859-1877. MR 2726101, https://doi.org/10.4007/annals.2010.172.1859
  • [14] K. Mori, Left invariant Einstein metrics on SU(n) that are not naturally reductive, master's thesis (in Japanese), Osaka University, 1994; English translation: Osaka University RPM 96-10 (preprint series) (1996).
  • [15] A. H. Mujtaba, Homogeneous Einstein metrics on $ SU(n)$, J. Geom. Phys. 62 (2012), no. 5, 976-980. MR 2901841, https://doi.org/10.1016/j.geomphys.2012.01.011
  • [16] Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskiĭ, Geometry of homogeneous Riemannian manifolds, Sovrem. Mat. Prilozh. 37, Geometriya (2006), 101-178 (Russian); English transl., J. Math. Sci. (N. Y.) 146 (2007), no. 6, 6313-6390. MR 2568572, https://doi.org/10.1007/s10958-007-0472-z
  • [17] C. N. Pope, Homogeneous Einstein metrics on SO(n), arXiv:1001.2776 (2010).
  • [18] McKenzie Y. Wang, Einstein metrics from symmetry and bundle constructions, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., VI, Int. Press, Boston, MA, 1999, pp. 287-325. MR 1798614, https://doi.org/10.4310/SDG.2001.v6.n1.a11
  • [19] McKenzie Y. Wang and Wolfgang Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 563-633. MR 839687
  • [20] McKenzie Y. Wang and Wolfgang Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986), no. 1, 177-194. MR 830044, https://doi.org/10.1007/BF01388738
  • [21] Joseph A. Wolf, The goemetry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968), 59-148. MR 0223501

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C25, 53C35, 53C30

Retrieve articles in all journals with MSC (2010): 53C25, 53C35, 53C30


Additional Information

Zaili Yan
Affiliation: Department of Mathematics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China
Email: yanzaili@nbu.edu.cn

Shaoqiang Deng
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: dengsq@nankai.edu.cn

DOI: https://doi.org/10.1090/tran/7025
Keywords: Einstein metrics, compact Lie groups, naturally reductive metrics, standard homogeneous Einstein manifolds
Received by editor(s): October 15, 2015
Received by editor(s) in revised form: January 27, 2016
Published electronically: May 5, 2017
Additional Notes: The first author was supported by NSFC (Nos. 11626134, 11401425) and K. C. Wong Magna Fund in Ningbo University.
The second author was supported by NSFC (Nos. 11671212, 51535008) of China
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society