Advertisement

Mathematische Annalen

, Volume 328, Issue 4, pp 711–748 | Cite as

On the holonomy of connections with skew-symmetric torsion

  • Ilka Agricola
  • Thomas Friedrich
Article

Abstract

We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits 2-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.

Keywords

Manifold Riemannian Manifold Euclidian Space Integral Formula Parameter Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agricola, I.: Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory. Comm. Math. Phys. 232, 535–563 (2003)zbMATHGoogle Scholar
  2. 2.
    Agricola, I., Friedrich, Th.: Killing spinors in supergravity with 4-fluxes, Class. Quant. Grav. 20, 4707–4717 (2003)CrossRefGoogle Scholar
  3. 3.
    Agricola, I., Friedrich, Th.: The Casimir operator of a metric connection with totally skew-symmetric torsion, to appear in Jour. Geom. Phys.Google Scholar
  4. 4.
    Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds. Teubner-Texte zur Mathematik, Band 124, Teubner-Verlag Stuttgart/Leipzig, 1991Google Scholar
  5. 5.
    Belgun, F., Moroianu, A.: Nearly Kähler 6-manifolds with reduced holonomy, Ann. Global Anal. Geom. 19, 307–319 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bismut, J.M.: A local index theorem for non-Kählerian manifolds. Math. Ann. 284, 681–699 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Boyer, C.P., Galicki, K.: 3-Sasakian manifolds. In: Essays on Einstein manifolds, C. LeBrun and M. Wang, (eds.), International Press, 1999Google Scholar
  8. 8.
    Cabrera, F.M., Monar, M.D., Swann, A.F.: Classification of G 2-structures. J. Lond. Math. Soc. II. Ser. 53, 407–416 (1996)zbMATHGoogle Scholar
  9. 9.
    Cartan, E.: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. C. R. Ac. Sc. 174, 593–595 (1922)zbMATHGoogle Scholar
  10. 10.
    Cartan, E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Ann. Ec. Norm. Sup. 40, 325–412 (1923); et Ann. Ec. Norm. Sup. 41, 1–25 (1924)zbMATHGoogle Scholar
  11. 11.
    Cartan, E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée (deuxième partie). Ann. Ec. Norm. Sup. 42, 17–88 (1925)zbMATHGoogle Scholar
  12. 12.
    Cartan, E.: Les récentes généralisations de la notion d’espace. Bull. Sc. Math. 48, 294–320 (1924)zbMATHGoogle Scholar
  13. 13.
    Fernandez, M.: A classification of Riemannian manifolds with structure group Spin(7). Ann. Mat. Pura Appl. 143, 101–122 (1986)zbMATHGoogle Scholar
  14. 14.
    Fernandez, M., Gray, A.: Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 132, 19–45 (1982)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Friedrich, Th.: Der erste Eigenwert des Dirac Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97, 117–146 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Friedrich, Th.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics 25, AMS, Privence, Rhode Island, 2000Google Scholar
  17. 17.
    Friedrich, Th.: On types of non-integrable geometries. Suppl. Rend. Circ. Mat. di Palermo, II Ser. 72, 99–113 (2003)Google Scholar
  18. 18.
    Friedrich, Th., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–336 (2002)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Friedrich, Th., Ivanov, S.: Almost contact manifolds, connections with torsion and parallel spinors. J. Reine Angew. Math. 559, 217–236 (2003)zbMATHGoogle Scholar
  20. 20.
    Friedrich, Th., Ivanov, S.: Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds. Journ. Geom. Phys. 48, 1–11 (2003)CrossRefGoogle Scholar
  21. 21.
    Friedrich, Th., Kath, I.: Compact seven-dimensional manifolds with Killing spinors. Comm. Math. Phys. 133, 543–561 (1990)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Friedrich, Th., Kath, I., Moroianu, A., Semmelmann, U.: On nearly parallel G2-structures. J. Geom. Phys. 23, 256–286 (1997)CrossRefGoogle Scholar
  23. 23.
    Hitchin, N.: Stable forms and special metrics, Contemp. Math. 288, 70–89 (2001)zbMATHGoogle Scholar
  24. 24.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry I. Wiley, 1963Google Scholar
  25. 25.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry II. Wiley, 1969Google Scholar
  26. 26.
    Lichnerowicz, A.: Spin manifolds, Killing spinors and universality of the Hijazi inequality. Lett. Math. Phys. 13, 331–344 (1987)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Trautman, A.: On the structure of the Einstein-Cartan equations. Symp. Math. 12, 139–162 (1973)zbMATHGoogle Scholar
  28. 28.
    Tricerri, F., Vanhecke, L.: Homogeneous structures on Riemannian manifolds. London Math. Soc. Lecture Notes Series 83, Cambridge Univ. Press, 1983Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für MathematikBerlinGermany

Personalised recommendations