Transformation Groups

, Volume 23, Issue 1, pp 257–270 | Cite as




In this paper, we conjecture an obstruction for arbitrary Lie groups to admit left-invariant Ricci solitons, from the viewpoint of isometric actions on noncompact symmetric spaces. We also construct examples of Lie groups that affirm the conjecture, in any dimension greater than two.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Д. В. Алексеевский, Б. Н. Кимельфельд, Строение однородных римановых пространств с нулевой кривизной Риччи, Функц. анализ и его прил. 9 (1975), въш. 2, 5–11. Engl. transl.: D. V. Alekseevskiĭ, B. N. Kimel′fel′d, Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funkcional Anal. Appl. 9 (1975), no. 2, 97–102.Google Scholar
  2. [2]
    R. M. Arroyo, Filiform nilsolitons of dimension 8, Rocky Mountain J. Math. 41 (2011), 1025–1043.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R. M. Arroyo, R. Lafuente, Homogeneous Ricci solitons in low dimensions, Int. Math. Res. Not. IMRN (2015), no. 13, 4901–4932.Google Scholar
  4. [4]
    J. Berndt, J. C. Díaz-Ramos, H. Tamaru, Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Differential Geom. 86 (2010), 191–235.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J. Berndt, H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom. 63 (2003), 1–40.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. Berndt, H. Tamaru, Cohomogeneity one actions on symmetric spaces of noncompact type, J. Reine Angew. Math. 683 (2013), 129–159.MathSciNetMATHGoogle Scholar
  7. [7]
    E. A. Fernández-Culma, Classification of 7-dimensional Einstein nilradicals, Transform. Groups 17 (2012), no. 3, 639–656.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    J. Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), 279–352.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    T. Hashinaga, H. Tamaru, Three-dimensional solsolitons and minimality of the corresponding submanifolds, Internat. J. Math. 28 (2017), no. 6, 1750048, 31 pp.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    T. Hashinaga, H. Tamaru, K. Terada, Milnor-type theorems for left-invariant Riemannian metrics on Lie groups, J. Math. Soc. Japan 68 (2016), no. 2, 669–684.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Jablonski, Moduli of Einstein and non-Einstein nilradicals, Geom. Dedicata 152 (2011), 63–84.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    M. Jablonski, Homogeneous Ricci solitons are algebraic, Geom. Topol. 18 (2014), no. 4, 2477–2486.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Jablonski, Homogeneous Ricci solitons, J. Reine Angew. Math. 699 (2015), 159–182.MathSciNetMATHGoogle Scholar
  14. [14]
    M. Jablonski, Strongly solvable spaces, Duke Math. J. 164 (2015), no. 2, 361–402.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    M. Jablonski, P. Petersen, A step towards the Alekseevskii Conjecture, Math. Ann. 368 (2017), no. 1–2, 197–212.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Kubo, H. Tamaru, A sufficient condition for congruency of orbits of Lie groups and some applications, Geom. Dedicata 167 (2013), 233–238.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    H. Kodama, A. Takahara, H. Tamaru, The space of left-invariant metrics on a Lie group up to isometry and scaling, Manuscripta Math. 135 (2011), 229–243.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. Lauret, Degenerations of Lie algebras and geometry of Lie groups, Differential Geom. Appl. 18 (2003), 177–194.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    J. Lauret, Einstein solvmanifolds and nilsolitons, Contemp. Math. 491 (2009), 1–35.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011), 1–21.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    R. Lafuente, J. Lauret, Structure of homogeneous Ricci solitons and the Alekseevskii conjecture, J. Differential Geom. 98 (2014), no. 2, 315–347.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    J. Lauret, C. Will, Einstein solvmanifolds: existence and non-existence questions, Math. Ann. 350 (2011), 199–225.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), 293–329.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Y. Nikolayevsky, Einstein solvmanifolds with a simple Einstein derivation, Geom. Dedicata 135 (2008), 87–102.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Y. Nikolayevsky, Einstein solvmanifolds with free nilradical, Ann. Global Anal. Geom. 33 (2008), 71–87.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Y. Nikolayevsky, Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer. Math. Soc. 363 (2011), 3935–3958.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Y. Nikolayevsky, Einstein solvmanifolds attached to two-step nilradicals, Math. Z. 272 (2012), 675–695.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    C. Will, A curve of nilpotent Lie algebras which are not Einstein nilradicals, Monatsh. Math. 159 (2010), 425–437.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    C. Will, The space of solvsolitons in low dimensions, Ann. Global Anal. Geom. 40 (2011), 291–309.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan

Personalised recommendations