manuscripta mathematica

, Volume 156, Issue 1–2, pp 127–136 | Cite as

Non-naturally reductive Einstein metrics on \(\mathrm {SO}(n)\)

Article
  • 15 Downloads

Abstract

In this article, we prove that every compact simple Lie group \({\mathrm S}{\mathrm O}(n)\) for \(n\ge 10\) admits at least \(2\left( [\frac{n-1}{3}]-2\right) \) non-naturally reductive left-invariant Einstein metrics.

Mathematics Subject Classification

53C25 53C30 53C35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arvanitoyeorgos, A., Mori, K., Sakane, Y.: Einstein metrics on compact Lie groups which are not naturally reductive. Geom. Dedicata 160(1), 261–285 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arvanitoyeorgos, A., Sakane, Y., Statha, M.: New Einstein metrics on the Lie group \(SO(n)\) which are not naturally reductive. Geom. Imaging Comput. 2(2), 77–108 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1986)MATHGoogle Scholar
  4. 4.
    Bohm, C.: Homogeneous Einstein metrics and simplicial complexes. J. Differ. Geom. 67(1), 74–165 (2004)MathSciNetMATHGoogle Scholar
  5. 5.
    Bohm, C., Wang, M., Ziller, W.: A variational approach for compact homogeneous Einstein manifolds. Geom. Func. Anal. 14(4), 681–733 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, H., Chen, Z.: Notes on “Einstein metrics on compact simple Lie groups attached to standard triples". arXiv:1701.01713v1 (2017)
  7. 7.
    Chen, Z., Kang, Y., Liang, K.: Invariant Einstein metrics on three-locally-symmetric spaces. Commun. Anal. Geom. 24(4), 769–792 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, Z., Liang, K.: Non-naturally reductive Einstein metrics on the compact simple Lie group \(F_4\). Ann. Glob. Anal. Geom. 46, 103–115 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chrysikos, I., Sakane, Y.: Non-naturally reductive Einstein metrics on exceptional Lie groups. J. Geom. Phys. 116, 152–186 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D’ Atri, J.E., Ziller, W.: Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups, Memoirs American Mathematical Society, vol. 215, American Mathematical Society, Providence (1979)Google Scholar
  11. 11.
    Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lauret, J.: Einstein solvmanifolds are standard. Ann. Math. (2) 172(3), 1859–1877 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mori, K.: Left Invariant Einstein Metrics on \(SU(n)\) that are not naturally reductive, Master Thesis (in Japanese) Osaka University 1994, English Translation: Osaka University RPM 96010 (preprint series) (1996)Google Scholar
  14. 14.
    Park, J.S., Sakane, Y.: Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20(1), 51–61 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wang, M.: Einstein metrics from symmetry and bundle constructions. In: Lebrun, C., Wang, M. (eds.) Surveys in Differential Geometry: Essays on Einstein Manifolds. Survveys in Differential Geometry VI, Int. Press, Boston, Ma (1999)Google Scholar
  16. 16.
    Wang, M.: Einstein metrics from symmetry and bundle constructions: A sequel. In: Shen, Y., Shen, Z., Yau, S.T. (eds.) Differential Geometry: Under the Influence of S.-S. Chern, Advanced Lectures in Mathematics, vol. 22, International Press, pp. 253–309 (2012)Google Scholar
  17. 17.
    Wang, M., Ziller, W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yan, Z., Deng, S.: Einstein metrics on compact simple Lie groups attached to standard triples, Trans. Am. Math. Soc. (2017). doi: 10.1090/tran/7025

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinPeople’s Republic of China

Personalised recommendations