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On solvable Lie groups of negative Ricci curvature

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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.

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References

  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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

  6. Burde, D.: Degenerations of nilpotent Lie algebras. J. Lie Theory 9, 193–202 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Chevalley, C.: Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind. 1152. Hermann & Cie, Paris (1951)

    Google Scholar 

  8. Dotti Miatello, I., Leite, M.L., Miatello, R.: Negative Ricci curvature on complex simple Lie groups. J. Geom. Dedicata 17, 207–218 (1984)

    MATH  MathSciNet  Google Scholar 

  9. Dotti Miatello, I.: Ricci curvature of left invariant metrics on solvable unimodular Lie groups. Math. Z. 180, 257–263 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gantmacher, F.R.: Applications of the Theory of Matrices. Interscience Publishers Ltd., London (1959)

    MATH  Google Scholar 

  11. Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Heintze, E.: On homogeneous manifolds of negative curvature. Mat. Ann. 211, 23–34 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Mathematics, 80. Academic Press, Inc, New York (1978)

    Google Scholar 

  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)

    MATH  Google Scholar 

  15. Laub, A.J., Meyer, K.: Canonical forms for symplectic and Hamiltonian matrices. Celest. Mech. 9, 213–238 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350, 199–225 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, M.Y., Wang, L.: A criterion for stability of matrices. J. Math. Anal. Appl. 225, 249–264 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nesterenko, M., Popovych, R.: Contractions of low-dimensional Lie algebras. J. Math. Phys. 47(12), 123515 (2006)

  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)

  21. Richardson, R.W.: Conjugacy classes of \(n\)-tuples in Lie algebras and algebraic groups. Duke Math. J. 57, 1–35 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rubin, J.L., Winternitz, P.: Solvable Lie algebras with Heisenberg ideals. J. Phys. A 26, 1123–1138 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 2(96), 277–295 (1972)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Melbourne, 3086, Australia

    Y. Nikolayevsky

  2. South Mathematical Institute of VSC RAS, 22 Markus St, 362027, Vladikavkaz, Russia

    Yu. G. Nikonorov

Authors
  1. Y. Nikolayevsky
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  2. Yu. G. Nikonorov
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Corresponding author

Correspondence to Y. Nikolayevsky.

Additional information

The first author is partially supported by ARC Discovery Grant DP130103485. The second author is supported in part by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-921.2012.1) and by Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009–2013 (Agreement No. 8206, Application No. 2012-1.1-12-000-1003-014).

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Nikolayevsky, Y., Nikonorov, Y.G. On solvable Lie groups of negative Ricci curvature. Math. Z. 280, 1–16 (2015). https://doi.org/10.1007/s00209-015-1410-2

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