Advertisement

Mathematische Annalen

, Volume 367, Issue 1–2, pp 283–309 | Cite as

The Alekseevskii conjecture in low dimensions

  • Romina M. Arroyo
  • Ramiro A. LafuenteEmail author
Article

Abstract

The long-standing Alekseevskii conjecture states that a connected homogeneous Einstein space G / K of negative scalar curvature must be diffeomorphic to \({\mathbb R}^n\). This was known to be true only in dimensions up to 5, and in dimension 6 for non-semisimple G. In this work we prove that this is also the case in dimensions up to 10 when G is not semisimple. For arbitrary G, besides 5 possible exceptions, we show that the conjecture holds up to dimension 8.

Notes

Acknowledgments

It is our pleasure to thank Jorge Lauret for fruitful discussions, and Christoph Böhm for providing useful comments on a draft version of this article. Part of this research was carried out while the first author was a visitor at McMaster University. She is very grateful to the Department of Mathematics, the Geometry and Topology group and especially to McKenzie Wang for his kindness and hospitality.

References

  1. 1.
    Alekseevskiĭ, D., Cortés, V.: Isometry groups of homogeneous quaternionic Kähler manifolds. J. Geom. Anal. 9(4), 513–545 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseevskiĭ, D., Dotti, I., Ferraris, C.: Homogeneous Ricci positive 5-manifolds. Pacific J. Math. 175, 1–12 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alekseevskiĭ, D., Kimel’fel’d, B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funktional. Anal. i Prilov Zen. 9(2), 5–11 (1975)Google Scholar
  4. 4.
    Alekseevsky, D.: Homogeneous Lorentzian manifolds of a semisimple group. J. Geom. Phys. 62(3), 631–645 (2012)Google Scholar
  5. 5.
    Aloff, S., Wallach, N.R.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)Google Scholar
  6. 6.
    Arroyo, R.M., Lafuente, R.: Homogeneous Ricci solitons in low dimensions. Int. Math. Res. Not 2015(13), 4901–4932 (2015)Google Scholar
  7. 7.
    Besse, A.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10. Springer-Verlag, Berlin (1987)Google Scholar
  8. 8.
    Bochner, S.: Curvature and Betti numbers. Ann. Math. 49(2), 379–390 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Böhm, C.: On the long time behavior of homogeneous Ricci flows. Comment. Math. Helv. 90, 543–571 (2015)Google Scholar
  10. 10.
    Böhm, C., Kerr, M.M.: Low-dimensional homogeneous Einstein manifolds. Trans. Am. Math. Soc. 358(4), 1455–1468 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cartan, E.: Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple. Ann. Sci. École Norm. Sup. 44(3), 345–467 (1927)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dotti Miatello, I.: Transitive group actions and ricci curvature properties. Michigan Math. J 35(3), 427–434 (1988)Google Scholar
  13. 13.
    Fernandez-Culma, E.: Classification of Nilsoliton metrics in dimension seven. J. Geom. Phys. 86, 164–179 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    He, C., Petersen, P., Wylie, W.: Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons. J. Reine Angew. Math (2014) (in press)Google Scholar
  15. 15.
    Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133(2), 279–352 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Pure Appl. Math., vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1978)Google Scholar
  17. 17.
    Jablonski, M.: Homogeneous Ricci solitons are algebraic. Geom. Topol. 18(4), 2477–2486 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jablonski, M.: Homogeneous Ricci solitons. J. Reine Angew. Math. 699, 159–182 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Jablonski, M.: Strongly solvable spaces. Duke Math. J. 164(2), 361–402 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jablonski, M., Petersen, P.: A Step Towards the Alekseevskii Conjecture (2014). arXiv:1403.5037v2
  21. 21.
    Jensen, G.: Homogeneous Einstein spaces of dimension four. J. Differ. Geomy 3, 309–349 (1969)Google Scholar
  22. 22.
    Lafuente, R., Lauret, J.: Structure of homogeneous Ricci solitons and the Alekseevskii conjecture. J. Differ. Geom. 98(2), 315–347 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lauret, J.: Finding einstein solvmanifolds by a variational method. Math. Z. 241, 83–99 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lauret, J.: Einstein solvmanifolds and nilsolitons, new developments in Lie theory and geometry, pp. 1–35. Contemp. Math., vol 491, Amer. Math. Soc. (2009)Google Scholar
  25. 25.
    Lauret, J.: Ricci soliton solvmanifolds. J. Reine Angew. Math. 650, 1–21 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lauret J.: Personal Communication (2012)Google Scholar
  27. 27.
    LeBrun, C., Wang, M (eds.).: Surveys in differential geometry: essays on Einstein manifolds, Surveys in Differential Geometry, VI, International Press, Boston, MA, 1999, Lectures on geometry and topology, sponsored by Lehigh University’s Journal of Differential GeometryGoogle Scholar
  28. 28.
    Milnor, J.: Curvatures of left-invariant metrics on lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nikitenko, E,V., Nikonorov, Y.: Six-dimensional Einstein solvmanifolds. Transl. Mat. Tr. 8(1), 71–121 (2005); mr1955023, Siberian Adv. Math. 16(1), 66–112 (2006)Google Scholar
  30. 30.
    Nikonorov, Y.: On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces. Sibirsk. Mat. Zh. 41(2), 421–429 (2000). ivMathSciNetzbMATHGoogle Scholar
  31. 31.
    Nikonorov, Y.: Compact homogeneous Einstein 7-manifolds. Geom. Dedicata 109(1), 7–30 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nikonorov, Y.: Noncompact homogeneous Einstein 5-manifolds. Geom. Dedicata 113, 107–143 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nikonorov, Y., Rodionov, E.D.: Compact homogeneous Einstein 6-manifolds. Diff. Geom. Appl. 19(3), 369–378 (2003)Google Scholar
  34. 34.
    Richardson, R.W., Slodowy, P.: Minimum vectors for real reductive algebraic groups. J. London Math. Soc 42(3), 409–429 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sparks, J.: Sasaki-Einstein manifolds, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics pp. 265–324. Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA (2011)Google Scholar
  36. 36.
    Spiro, A.: A remark on locally homogeneous Riemannian spaces. Results Math. 24(3–4), 318–325 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tricerri, F.: Locally homogeneous Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino. 50 (1992), (4), 411–426 (1993), Differential geometry (Turin, 1992)Google Scholar
  38. 38.
    Wang, M.Y.: Some examples of homogeneous Einstein manifolds in dimension seven. Duke Math. J. 49(1), 23–28 (1982)Google Scholar
  39. 39.
    Wang, M.Y.: Einstein metrics from symmetry and bundle constructions: a sequel. Differential Geometry: Under the Influence of S.-S. Chern, Advanced Lectures in Mathematics, pp. 253–309, vol. 22, Higher Education Press/International Press, Beijing-Boston (2012)Google Scholar
  40. 40.
    Will, C.: Rank-one einstein solvmanifolds of dimension 7. Diff. Geom. Appl. 19, 307–318 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.FaMAF and CIEMUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Mathematisches InstitutUniversität MünsterMünsterGermany

Personalised recommendations