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Annals of Global Analysis and Geometry

, Volume 33, Issue 1, pp 71–87 | Cite as

Einstein solvmanifolds with free nilradical

  • Yuri NikolayevskyEmail author
Original Paper

Abstract

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the Abelian and the two-step ones, there are only six others: \({\mathfrak{f}}(2,3), {\mathfrak{f}}(2,4), {\mathfrak{f}}(2,5), {\mathfrak{f}}(3,3), {\mathfrak{f}}(4,3), {\mathfrak{f}}(5,3) (here {\mathfrak{f}}(m,p)\) is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradical obtained in the paper.

Keywords

Einstein nilradical Free Lie algebra 

Mathematics Subject Classification (2000)

53C30 53C25 

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References

  1. 1.
    Alekseevskii D.V. (1975). Classification of quaternionic spaces with transitive solvable group of motions, Math. USSR – Izv. 9: 297–339 CrossRefGoogle Scholar
  2. 2.
    Alekseevskii D.V. (1975). Homogeneous Riemannian spaces of negative curvature. Math. USSR. Sb. 25: 87–109 CrossRefGoogle Scholar
  3. 3.
    Alekseevskii D.V., Kimel’fel’d B.N. (1975). Structure of homogeneous Riemannian spaces with zero Ricci curvature. Functional Anal. Appl. 9: 97–102 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Besse A. (1987). Einstein Manifolds. Springer, Berlin zbMATHGoogle Scholar
  5. 5.
    Bourbaki N. (1971). Éléments de Mathématique Groupes et Algèbres de Lie, Ch. 2. Hermann, Paris Google Scholar
  6. 6.
    Böhm C., Wang M., Ziller M. (2004). A variational approach for compact homogeneous Einstein manifolds. GAFA 14: 681–733 zbMATHGoogle Scholar
  7. 7.
    Dotti Miatello I. (1982). Ricci curvature of left-invariant metrics on solvable unimodular Lie groups. Math. Z. 180: 257–263 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fanaï H.-R. (2002). Variétés homogènes d’Einstein de courbure scalaire négative: construction à l’aide de certains modules de Clifford. Geom. Dedicata 93: 77–87 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gordon C., Kerr M. (2001). New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19: 75–101 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Heber J. (1998). Noncompact homogeneous Einstein spaces. Invent. Math. 133: 279–352 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lauret J. (2001). Ricci soliton homogeneous nilmanifolds. Math. Ann. 319: 715–733 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lauret J. (2001). Standard Einstein solvmanifolds as critical points. Quart. J. Math. 52: 463–470 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lauret J. (2002). Finding Einstein solvmanifolds by a variational method. Math. Z. 241: 83–99 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lauret, J.: Minimal metrics on nilmanifolds. Diff. Geom. Appl. Proc. Conf. Prague September 2004, pp. 77–94 (2005)Google Scholar
  15. 15.
    Lauret, J.: Einstein solvmanifolds are standard, preprint 2007, arXiv: math/0703472Google Scholar
  16. 16.
    Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions, preprint 2006, arXiv: math.DG/0602502Google Scholar
  17. 17.
    Nikolayevsky, Y.: Nilradicals of Einstein solvmanifolds. preprint 2006, arXiv: math.DG/0612117Google Scholar
  18. 18.
    Nikonorov Yu.G. (2005). Noncompact homogeneous Einstein 5-manifolds. Geom. Dedicata 113: 107–143 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Nikitenko E.V., Nikonorov Yu.G. (2006). Six-dimensional Einstein solvmanifolds Siberian. Adv. Math. 16: 66–112 MathSciNetGoogle Scholar
  20. 20.
    Payne, T.: The existence of soliton metrics for nilpotent Lie groups, preprint 2005Google Scholar
  21. 21.
    Reutenauer C. (1993). Free Lie Algebras London Mathematical Society Monographs, New series, Vol. 7. Clarendon Press, Oxford Google Scholar
  22. 22.
    Schueth D. (2004). On the “standard” condition for noncompact homogeneous Einstein spaces. Geom. Dedicata 105: 77–83 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Tamaru, H.: Noncompact homogeneous Einstein manifolds attached to graded Lie algebras, preprint 2006, arXiv: math.DG/0610675Google Scholar
  24. 24.
    Wang M., Ziller W. (1986). Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84: 177–194 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsLa Trobe UniversityMelbourneAustralia

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