Mathematische Zeitschrift

, Volume 252, Issue 4, pp 825–848 | Cite as

Conformally parallel G2 structures on a class of solvmanifolds



Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G2 structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G2. In the process we also find a new metric with exceptional holonomy.

Mathematics Subject Classification (2000)

Primary 53C10 Secondary 53C25 53C29 22E25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbena, E., Garbiero, S., Salamon, S.: Almost Hermitian geometry on six dimensional nilmanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(1), 147–170 (2001)Google Scholar
  2. 2.
    Alekseevskii, D.V., Kimel'fel'd, B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funkcional. Anal. i PriloŽen. 9(2), 5–11 (1975)Google Scholar
  3. 3.
    Apostolov, V., Salamon, S.: Kähler reduction of metrics with holonomy G 2. Comm. Math. Phys. 246(1), 43–61 (2004)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Becker, M., Dasgupta, K., Knauf, A., Tatar, R.: Geometric transitions, flops and non-Kähler manifolds: I. Nuclear Phys. B 702(1–2), 207–268 (2004)Google Scholar
  5. 5.
    Bonan, E.: Sur des variétés riemanniennes à groupe d'holonmie G 2 ou Spin(7). C. R. Acad. Sci. Paris, 262, 127–129 (1966)MathSciNetGoogle Scholar
  6. 6.
    Bor, G., Hernández Lamoneda, L.: Bochner formulae for orthogonal G-structures on compact manifolds. Differential Geom. Appl. 15(3), 265–286 (2001)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bryant, R.L.: Metrics with exceptional holonomy. Ann. of Math. 126, 525–576 (1987)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bryant, R.L.: Some remarks on G 2-structures. May 2003, eprint arXiv:math.DG/0305124Google Scholar
  9. 9.
    Bryant, R.L., Salamon, S.M.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cabrera, F.M., Monar, M.D., Swann, A.F.: Classification of G 2-structures. J. London Math. Soc. 53, 407–416 (1996)MATHMathSciNetGoogle Scholar
  11. 11.
    Chiossi, S.G., Salamon, S.: The intrinsic torsion of SU(3) and G 2 structures. In: Gil-Medrano, O., Miquel, V. (eds.), Differential geometry, Valencia, 2001, World Sci. Publishing, River Edge, NJ, 2002, pp. 115–133Google Scholar
  12. 12.
    Cleyton, R., Ivanov, S.: On the geometry of closed G 2-structures. June 2003, eprint arXiv:math.DG/0306362Google Scholar
  13. 13.
    Cleyton, R., Swann, A.: Strong G 2 manifolds with cohomogeneity-one actions of simple Lie groups. In preparationGoogle Scholar
  14. 14.
    Dotti, I.: Ricci curvature of left invariant metrics on solvable unimodular Lie groups. Math. Z. 180(2), 257–263 (1982)MathSciNetGoogle Scholar
  15. 15.
    Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. (4) 132(1982), 19–45 (1983)Google Scholar
  16. 16.
    Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)MathSciNetGoogle Scholar
  17. 17.
    Gibbons, G.W., Lü, H., Pope, C.N., Stelle, K.S.: Supersymmetric domain walls from metrics of special holonomy. Nuclear Phys. B 623(1-2), 3–46 (2002)Google Scholar
  18. 18.
    Goze, M., Khakimdjanov, Y.: Nilpotent Lie algebras. Mathematics and its Applications, vol. 361, Kluwer Academic Publishers Group, Dordrecht, 1996Google Scholar
  19. 19.
    Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror symmetry in generalized Calabi-Yau compactifications. Nuclear Phys. B 654(1-2), 61–113 (2003)Google Scholar
  20. 20.
    Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133(2), 279–352 (1998)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hitchin, N.J.: Stable forms and special metrics. Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001, pp. 70–89Google Scholar
  22. 22.
    Joyce, D.: Compact Riemannian 7-manifolds with holonomy G 2: I, II. J. Differential Geom. 43, 291–328, 329–375 (1996)MATHMathSciNetGoogle Scholar
  23. 23.
    Joyce, D.D.: Compact manifolds with special holonomy. Oxford University Press, Oxford, 2000Google Scholar
  24. 24.
    Ketsetzis, G., Salamon, S.: Complex structures on the Iwasawa manifold. Adv. in Geometry 4, 165–179 (2004)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003)MATHMathSciNetGoogle Scholar
  26. 26.
    Lauret, J.: Standard Einstein solvmanifolds as critical points. Q. J. Math. 52(4), 463–470 (2001)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Magnin, L.: Sur les algèbres de Lie nilpotentes de dimension ≤7. J. Geom. Phys. 3(1), 119–144 (1986)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Mal'cev, A.I.: On a class of homogeneous spaces. Amer. Math. Soc. Translation 1951(39), 33 (1951), originally appearing in Izv. Akad. Nauk. SSSR. Ser. Mat. 13, 9–32 (1949)MathSciNetGoogle Scholar
  29. 29.
    Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. 59, (2) 531–538 (1954)Google Scholar
  30. 30.
    Salamon, S.M.: Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics, vol. 201, Longman, Harlow, 1989Google Scholar
  31. 31.
    Will, C.: Rank-one Einstein solvmanifolds of dimension 7. Differential Geom. Appl. 19(3), 307–318 (2003)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Wilson, E.N.: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12(3), 337–346 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Institut für Mathematik der Humboldt-UniversitätBerlinGermany
  2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

Personalised recommendations