Geometriae Dedicata

, Volume 113, Issue 1, pp 107–143 | Cite as

Noncompact Homogeneous Einstein 5-Manifolds



This article is devoted to the classification of noncompact homogeneous Einstein 5-manifolds. In particular, we prove that each noncompact homogeneous Einstein 5-manifolds is locally isometric to some standard Einstein solvmanifold


Riemannian manifolds solvmanifolds homogeneous spaces Einstein metrics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseevskii D., V. 1968Compact quaternionic spacesFunct. Anal. Appl21120Google Scholar
  2. 2.
    Alekseevskii D., V. 1970Quaternionic Riemannian spaces with a transitive reductive or solvable group of motionsFunct. Anal. Appl.46869CrossRefGoogle Scholar
  3. 3.
    Alekseevskii, D. V.: Conjugasy of polar factorizations of Lie groups, Mat. Sb. 84 (1971), 14–26; English translation: Math. USSR-Sb, 13 (1971), 12–24.Google Scholar
  4. 4.
    Alekseevskii, D. V.: Classification of quaternionic spaces with a transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1975), 93–117; English translation: Math. USSR-Izv. 9 (1975), 297–339.Google Scholar
  5. 5.
    Alekseevskii, D. V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb., 96 (1975), 93–117; English translation: Math. USSR-Sb. 25 (1975), 87–109.Google Scholar
  6. 6.
    Alekseevskii, D. V., Kimmel’fel’d, B. N. 1975Structure of homogeneous Riemannian spaces with zero Ricci curvatureFunct. Anal. Appl997102CrossRefGoogle Scholar
  7. 7.
    Alekseevskii, D. V., Dotti, I., Ferraris, C. 1996Homogeneous Ricci positive 5-manifoldsPacific J. Math.175112Google Scholar
  8. 8.
    Berard Bergery, L. 1978Sur la courbure des metriqes riemanniennes invariants des groupes de Lie et des espaces homogenesAnn. Sci. Ecole Norm. Sup.11543576Google Scholar
  9. 9.
    Berard Bergery, L.: Homogeneous Riemannian spaces of dimension 4, In: Gèomètrie riemannienne en dimension 4, Sèminaire Arthur Besse, Cedic, Paris, 1981.Google Scholar
  10. 10.
    Besse, A.. 1987Einstein ManifoldsSpringerNew YorkBerlin HeidelbergGoogle Scholar
  11. 11.
    Böhm, C. 2004Homogeneous einstein metrics and simplicial complexesJ. Differential Geom6779165Google Scholar
  12. 12.
    Böhm C.: and Kerr, M.: Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. (to appear).Google Scholar
  13. 13.
    Böhm, C., Wang, M., Ziller, W. 2004A variational approach for compact homogeneous Einstein manifoldsGeom. Funct. Anal14681733Google Scholar
  14. 14.
    Bourbaki, N. 1989Element of Mathematics: Lie Groups and Lie Algebras, Ch. 1-3SpringerBerlinGoogle Scholar
  15. 15.
    Cartan, È 1935Sur les domaines bornès homogènes de l’space de n variables complexesAbh Math Sem Univ Hamburg.11116162Google Scholar
  16. 16.
    Cortés, V. 1996Alekseevskian spacesDifferential Geom. Appl.6129168CrossRefGoogle Scholar
  17. 17.
    Dotti Miatello, I. 1988Transitive group actions and Ricci curvature propertiesMichigan Math J.35427434CrossRefGoogle Scholar
  18. 18.
    Dotti Miatello, I. 1982Ricci curvature of left-invariant metrics on solvable unimodular Lie groupsMath Zeit.180257263Google Scholar
  19. 19.
    Gantmahe, F.R. 1988The Theory of MatricesNaukaMoscowGoogle Scholar
  20. 20.
    Gindikin, S. G., Piatetskii-Shapiro, I. I. and Vinberg, E. B.: On the classification and canonical realization of bounded homogeneous domains. Trudy Moscov. Mat. Obshch., 12, (1963), 359–388; English translation: Trans. Moscow Math. Soc. 12 (1963), 404–437.Google Scholar
  21. 21.
    Gindikin, S. G., Piatetskii-Shapiro, I. I. and Vinberg, E. B.: Homogeneous Kähler manifolds, In: Geometry of Homogeneous Bounded Domains (C.I.M.E. 3, Ciclo, Urbino, 1967), Edizioni Cremoneze, Rome, 1968, pp 3–87.Google Scholar
  22. 22.
    Gordon C., S., Kerr, M. 2001New homogeneous metrics of negative Ricci curvatureAnn. Global Anal. Geom.1975101CrossRefGoogle Scholar
  23. 23.
    Gordon, C.S., Wilson, E.N. 1988Isometry groups of Riemannian solvmanifoldsTrans. Amer. Math. Soc307245269Google Scholar
  24. 24.
    Jensen, G. 1969Homogeneous Einstein spaces of dimension 4J. Differential Geom3309349Google Scholar
  25. 25.
    Jensen, G. 1971The scalar curvature of left invariant Riemannian metricsIndiana U. Math J2011251143CrossRefGoogle Scholar
  26. 26.
    Jensen, G. 1973Einstein metrics on principal fibre bundlesJ. Differential Geom8599614Google Scholar
  27. 27.
    Heber, J. 1998Noncompact homogeneous Einstein spacesInvent. Math133279352CrossRefGoogle Scholar
  28. 28.
    Helgason, S. 1978Differential Geometry, Lie Groups and Symmetric SpacesAcademic PressNew YorkGoogle Scholar
  29. 29.
    Kerr, M.: A deformation of quaternionic hyperbolic space, Proc. Amer Math Soc. (to appear).Google Scholar
  30. 30.
    Leite M., L., Miatello I., D. 1982Metrics of negative Ricci curvature on SL(n,R), n ≥ 3J. Differential Geom17635641Google Scholar
  31. 31.
    Nakajima, K. 1986On j-algebras and homogeneous Kähler manifoldsHokkaido Math. J15120Google Scholar
  32. 32.
    Nikonorov, Yu. G.: The scalar curvature functional and homogeneous Einsteinian metrics on Lie groups, Sib. Math. J. 39 (3) (1998), 504–509; translation from Sib. Mat. Zh. 39(3), (1998), 583–589.Google Scholar
  33. 33.
    Nikonorov, Yu. G.: On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces. Sib. Math. J. 41 (2) (2000), 349–356; translation from Sib. Mat. Zh. 41 (2) (2000), 421–429.Google Scholar
  34. 34.
    Nikonorov, Yu.G., Rodionov, E.D. 2003Compact homogeneous Einstein 6-manifoldsDifferential Geom. Appl19369378CrossRefGoogle Scholar
  35. 35.
    Nikonorov, Yu.G. 2004Compact homogeneous Einstein 7-manifoldsGeom. Dedicata.109730CrossRefMathSciNetGoogle Scholar
  36. 36.
    Piatetskii-Shapiro, I. I.: On the classification of bounded homogeneous domains in n-dimensional complex space, Dokl. Akad. Nauk SSSR 141 (1961), 316–319; English translation: Soviet Math. Dokl. 2 (1962), 1460–1463.Google Scholar
  37. 37.
    Piatetskii-Shapiro, I. I.: The structure of j-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1966), 453–484; English translation: Amer. Math. Soc. Transl. (2) 55 (1966), 207–241.Google Scholar
  38. 38.
    Schueth, D. 2004On the ’standard’ condition for noncompact homogeneous Einstein spacesGeom. Dedicata.1057783CrossRefGoogle Scholar
  39. 39.
    Vinberg, E. B. and Gindikin, S. G.: Kählerian manifolds admitting a transitive solvable automorphism group. Mat. Sb. 74 (1967), 357–377; English translation: Math. USSR Sb. 3 (1967), 333–352.Google Scholar
  40. 40.
    Vinberg, E. B.: Gorbatsevich, V. V. and Onishchik, A. L.: Structure of Lie Groups and Algebras, Itogi Nauki i Tekh., Ser. Probl. Mat., Fundam Napravleniya, 41 (1990).Google Scholar
  41. 41.
    Wang, M.: Einstein metrics from symmetry and bundle constructions, Surveys in Differential Geometry: Essays on Einstein Manifolds. Lectures on Geom. Topology, sponsored by Lehigh University’s J. Differential Geom. (Surv. Differential Geom. Suppl. J. Differential Geom. V.6), International Press, Cambridge, 1999. pp.287-325.Google Scholar
  42. 42.
    Wang, M., Ziller, W. 1986Existence and non-existence of homogeneous Einstein metricsInvent. Math84177194CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Rubtsovsk Industrial instituteRubtsovskRussia

Personalised recommendations