Skip to main content
SpringerLink
Log in

Einstein solvmanifolds attached to two-step nilradicals

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra which can serve as the nilradical of an Einstein metric solvable Lie algebra is called an Einstein nilradical. We give a classification of two-step nilpotent Einstein nilradicals with two-dimensional center. Informally, the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also show that the dual to a two-step Einstein nilradical is not in general an Einstein nilradical.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseevskii D.V.: Classification of quaternionic spaces with transitive solvable group of motions. Math. USSR. Izv. 9, 297–339 (1975)

    Article  Google Scholar 

  2. Alekseevskii D.V.: Homogeneous Riemannian spaces of negative curvature. Math. USSR. Sb. 25, 87–109 (1975)

    Article  Google Scholar 

  3. Alekseevskii D.V., Kimel’fel’d B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funct. Anal. Appl. 9, 97–102 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belitskii G., Lipyanski R., Sergeichuk V.: Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild. Linear Algebra Appl. 407, 249–262 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Besse A.: Einstein Manifolds. Springer, Berlin (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Eberlein P.: Geometry of 2-step nilpotent Lie groups. Modern Dynamical Systems, pp. 67–101. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  8. Eberlein, P.: Riemannian 2-step nilmanifolds with prescribed Ricci tensor. Geometric and Probabilistic Structures in Dynamics. Contemporary Mathematics, vol. 469, pp. 167–195. American Mathematical Soceity, Providence (2008)

  9. Galitski L., Timashev D.: On classification of metabelian Lie algebras. J. Lie Theory 9, 125–156 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Gantmacher F.R.: The Theory of Matrices. vol. 1, 2. Chelsea Publishing Co., New York (1959)

    Google Scholar 

  11. Gauger M.: On the classification of metabelian Lie algebras. Trans. Am. Math. Soc. 179, 293–329 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gordon C., Kerr M.: New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19, 75–101 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Jablonski, M.: Real Geometric Invariant Theory and Ricci soliton metrics on two-step nilmanifolds. Thesis (2008)

  15. Lancaster P., Rodman L.: Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence. Linear Algebra Appl. 406, 1–76 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lauret J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241, 83–99 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lauret J.: Einstein solvmanifolds are standard. Ann. Math. 172(2), 1859–1877 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lauret J.: Einstein solvmanifolds and nilsolitons. New developments in Lie theory and geometry Contemporary Mathematics, vol. 491, pp. 1–35. American Mathematical Soceity, Providence (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Nikolayevsky, Y.: Nilradicals of Einstein solvmanifolds. arXiv: math.DG/0612117 (Preprint, 2006)

  22. Nikolayevsky Y.: Einstein solmanifolds with free nilradical. Ann. Global Anal. Geom. 33, 71–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nikolayevsky Y.: Einstein solvmanifolds with a simple Einstein derivation. Geom. Dedicata 135, 87–102 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nikolayevsky Y.: Einstein solvmanifolds and the pre-Einstein derivation. Trans. Am. Math. Soc. 363, 3935–3958 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Payne T.: The existence of soliton metrics for nilpotent Lie groups. Geom. Dedicata 145, 71–88 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tamaru H.: Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds. Math. Ann. 351, 51–66 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Will C.: Rank-one Einstein solvmanifolds of dimension 7. Differ. Geom. Appl. 19, 307–318 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Yuri Nikolayevsky

Authors
  1. Yuri Nikolayevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Nikolayevsky.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nikolayevsky, Y. Einstein solvmanifolds attached to two-step nilradicals. Math. Z. 272, 675–695 (2012). https://doi.org/10.1007/s00209-011-0954-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0954-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation