Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 303–311 | Cite as

Holonomy rigidity for Ricci-flat metrics

  • Bernd AmmannEmail author
  • Klaus Kröncke
  • Hartmut Weiss
  • Frederik Witt


On a closed connected oriented manifold M we study the space \(\mathcal {M}_\Vert (M)\) of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space \(\mathcal {M}_\Vert (M)\) is a smooth submanifold of the space of all metrics and its premoduli space is a smooth finite-dimensional manifold. The holonomy group is locally constant on \(\mathcal {M}_\Vert (M)\). If M is spin, then the dimension of the space of parallel spinors is a locally constant function on \(\mathcal {M}_\Vert (M)\).



We thank X. Dai for discussions about the history of the subject and H.-J. Hein for some discussions related to Tian-Todorov theory.


  1. 1.
    Adams, F.: Lectures on exceptional Lie groups. In: Mahmud, Z., Mimura, M. (eds.) Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1996)Google Scholar
  2. 2.
    Ammann, B., Weiß, H., Witt, F.: A spinorial energy functional: critical points and gradient flow. Math. Ann. 365, 1559–1602 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Besse, A.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, New York (1987)Google Scholar
  4. 4.
    van Coevering, C.: Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds. J. Math. Soc. Jpn. 69, 53–91 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Charlap, L.S.: Bieberbach groups and flat manifolds, Universitext. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971/1972)Google Scholar
  7. 7.
    Dai, X., Wang, X., Wei, G.: On the stability of Riemannian manifold with parallel spinors. Invent. Math. 161(1), 151–176 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fischer, A., Wolf, J.: The structure of compact Ricci-flat Riemannian manifolds. J. Differ. Geom. 10, 277–288 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97, 117–146 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Friedrich, T.: Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25. AMS, Providence (2000)Google Scholar
  11. 11.
    Goto, R.: Moduli spaces of topological calibrations, Calabi-Yau, hyper-Kähler, \(G_2\) and Spin(7) structures. Int. J. Math. 15, 211–257 (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kröncke, K.: On infinitesimal Einstein deformations. Differ. Geom. Appl. 38, 41–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leistner, T.: Lorentzian manifolds with special holonomy and parallel spinors. [Preprint No. 486/SFB 288, see also Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 69, 131–159]
  15. 15.
    McInnes, B.: Methods of holonomy theory for Ricci-flat Riemannian manifolds. J. Math. Phys. 32, 888–896 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Montgomery, D., Zippin, L.: A theorem on Lie groups. Bull. Am. Math. Soc. 48, 448–452 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Moroianu, A., Semmelmann, U.: Parallel spinors and holonomy groups. J. Math. Phys. 41, 2395–2402 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nordström, J.: Ricci-flat deformations of metrics with exceptional holonomy. Bull. Lond. Math. Soc. 45(5), 1004–1018 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, M.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, M.: Preserving parallel spinors under metric deformations. Indiana Univ. Math. J. 40(3), 815–844 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, M.: On non-simply connected manifolds with non-trivial parallel spinors. Ann. Global Anal. Geom. 13(1), 31–42 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wilking, B.: On compact Riemannian manifolds with noncompact holonomy groups. J. Differ. Geom. 52(2), 223–257 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Bernd Ammann
    • 1
    Email author
  • Klaus Kröncke
    • 2
  • Hartmut Weiss
    • 3
  • Frederik Witt
    • 4
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Fachbereich MathematikUniversität HamburgHamburgGermany
  3. 3.Mathematisches SeminarUniversität KielKielGermany
  4. 4.Institut für Geometrie und TopologieUniversität StuttgartStuttgartGermany

Personalised recommendations