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Mathematische Zeitschrift

, Volume 280, Issue 1–2, pp 1–16 | Cite as

On solvable Lie groups of negative Ricci curvature

  • Y. Nikolayevsky
  • Yu. G. Nikonorov
Article

Abstract

We study solvable Lie groups which admit a left-invariant metric of strictly negative Ricci curvature. We obtain necessary and sufficient conditions of the existence of such a metric for Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform and discuss some open questions.

Keywords

Solvable Lie algebra Nilradical Negative Ricci curvature 

Mathematics Subject Classification

53C30 22E25 

References

  1. 1.
    Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Math. Sb. (N.S.) 96, 93–117 (1975) (Russian). English translation. In: Math. USSR-Sb. 25, 87–109 (1976)Google Scholar
  2. 2.
    Alekseevskii, D.V., Kimel’fel’d, B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funct. Anal. Appl. 9, 297–339 (1975)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bérard Bergery, L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pures Appl. 55, 47–67 (1976)MATHMathSciNetGoogle Scholar
  4. 4.
    Bérard Bergery, L.: Sur la courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes. Ann. Sci. École Norm. Sup. (4) 11, 543–576 (1978)MATHGoogle Scholar
  5. 5.
    Berestovskii, V.N.: Homogeneous Riemannian manifolds of positive Ricci curvature. Math. Zametki 58, 334–340 (1995) (Russian). English translation. In: Math. Notes 58, 905–909 (1995)Google Scholar
  6. 6.
    Burde, D.: Degenerations of nilpotent Lie algebras. J. Lie Theory 9, 193–202 (1999)MATHMathSciNetGoogle Scholar
  7. 7.
    Chevalley, C.: Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind. 1152. Hermann & Cie, Paris (1951)Google Scholar
  8. 8.
    Dotti Miatello, I., Leite, M.L., Miatello, R.: Negative Ricci curvature on complex simple Lie groups. J. Geom. Dedicata 17, 207–218 (1984)MATHMathSciNetGoogle Scholar
  9. 9.
    Dotti Miatello, I.: Ricci curvature of left invariant metrics on solvable unimodular Lie groups. Math. Z. 180, 257–263 (1982)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gantmacher, F.R.: Applications of the Theory of Matrices. Interscience Publishers Ltd., London (1959)MATHGoogle Scholar
  11. 11.
    Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Heintze, E.: On homogeneous manifolds of negative curvature. Mat. Ann. 211, 23–34 (1974)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Mathematics, 80. Academic Press, Inc, New York (1978)Google Scholar
  14. 14.
    Leite, M.L., Dotti Miatello, I.: Metrics of negative Ricci curvature on \(\text{ SL }(n,\mathbb{R}), n \ge 3\). J. Differ. Geom. 17, 635–641 (1982)MATHGoogle Scholar
  15. 15.
    Laub, A.J., Meyer, K.: Canonical forms for symplectic and Hamiltonian matrices. Celest. Mech. 9, 213–238 (1974)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350, 199–225 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Li, M.Y., Wang, L.: A criterion for stability of matrices. J. Math. Anal. Appl. 225, 249–264 (1998)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Nesterenko, M., Popovych, R.: Contractions of low-dimensional Lie algebras. J. Math. Phys. 47(12), 123515 (2006)Google Scholar
  20. 20.
    Nikitenko, E.V., Nikonorov, Yu.G.: Six-dimensional Einstein solvmanifolds. Mat. Tr. 8, 71–121 (2005) (Russian). English translation. In: Siberian Adv. Math. 16:1, 66–112 (2006)Google Scholar
  21. 21.
    Richardson, R.W.: Conjugacy classes of \(n\)-tuples in Lie algebras and algebraic groups. Duke Math. J. 57, 1–35 (1988)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rubin, J.L., Winternitz, P.: Solvable Lie algebras with Heisenberg ideals. J. Phys. A 26, 1123–1138 (1993)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Vergne, M.: Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes. Bull. Soc. Math. France 98, 81–116 (1970)MATHMathSciNetGoogle Scholar
  24. 24.
    Vinberg, È.B., Gorbatsevich, V.V., Onishchik, A.L.: Lie Groups and Lie Algebras, III. Structure of Lie Groups and Lie Algebras, Encyclopedia of Math. Sciences V. 41. Springer, Berlin (1994)Google Scholar
  25. 25.
    Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 2(96), 277–295 (1972)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia
  2. 2.South Mathematical Institute of VSC RASVladikavkazRussia

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