Transformation Groups

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

ON THE NONEXISTENCE OF LEFT-INVARIANT RICCI SOLITONS — A CONJECTURE AND EXAMPLES

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Abstract

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.

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References

  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

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