Skip to main content
SpringerLink
Log in

SO(3)-Structures on 8-manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We study Riemannian 8-manifolds with an infinitesimal action of SO(3) under which each tangent space breaks into irreducible spaces of dimensions 3 and 5. The relationship with quaternionic, almost product- and PSU-geometry is thoroughly explained using representation-theoretical arguments. This leads to the precise study of the intrinsic invariants upon which the structure depends, and also to the description of topological constraints for the existence of such manifolds. Finally, many examples are provided together with general recipes to build them.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agricola I.: The Srní lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42, 5–84 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Agricola I., Becker-Bender J., Friedrich Th.: On the topology and the geometry of SO(3)-manifolds. Ann. Global Anal. Geom. 40, 67–84 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atiyah M.F.: Vector Fields on Manifolds. Arbeitsgemeinschaft Forsch, Nordrhein-Westfalen Heft 200 (1970)

    Book  MATH  Google Scholar 

  4. Berger M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)

    MathSciNet  MATH  Google Scholar 

  5. Bobieński B., Nurowski P.: Irreducible SO(3)-geometries in dimension five. J. Reine Angew. Math. 605, 51–93 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Bonan, E.: Sur l’algèbre extérieure d’une variété presque hermitienne quaternionique. C. R. Acad. Sci. Paris Sér. I Math. 296(14), 601–602 (1983) and 295(2), 115–118 (1982)

  7. Čadek M., Vanžura J.: On Sp(2) and Sp(2)Sp(1)-structures in 8-dimensional vector bundles. Publ. Mat. 41(2), 383–401 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Čadek M., Vanžura J.: Almost quaternionic structures on eight-manifolds. Osaka J. Math. 35(1), 165–190 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Čadek M., Crabb M., Vanžura J.: Quaternionic structures. Topology Appl. 157(18), 2850–2863 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chiossi S.G., Fino A.: Nearly integrable SO(3) structures on five-dimensional Lie groups. J. Lie Theory 17(3), 539–562 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Friedrich Th.: On types of non-integrable geometries. Suppl. Rend. Circ. Mat. di Palermo Ser. II 71, 99–113 (2003)

    MathSciNet  Google Scholar 

  12. Gambioli A.: Latent quaternionic geometry. Tokyo J. Math. 31(1), 203–223 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gil-Medrano O.: Geometric properties of some classes of Riemannian almost-product manifolds. Rend. Circ. Mat. Palermo(2) 32(3), 315–329 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gray A., Wolf J.A.: Homogeneous spaces defined by Lie group automorphisms. I. J. Differential Geom. 2, 77–114 (1968)

    MathSciNet  MATH  Google Scholar 

  15. Gray A., Hervella L.: The sixteen classes of almost Hermitian manifolds. Ann. Mat. Pura Appl.(4) 123, 35–58 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heaps T.: Almost complex structures on eight- and ten-dimensional manifolds. Topology 9, 111–119 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hirzebruch, F.: Topological methods in algebraic geometry. 3rd edn. Grundlehren der Mathematischen Wissenschaften, Band 131 Springer, New York (1966)

  18. Hitchin, N.: Stable forms and special metrics. In: Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, 2000), pp. 70–89, Contemp. Math., 288, Amer. Math. Soc., Providence, RI (2001)

  19. Kraines V.Y.: Topology of quaternionic manifolds. Trans. Amer. Math. Soc. 122, 357–367 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. LeBrun, C., Wang, M.: (eds.) Surveys in Differential Geometry: Essays on Einstein Manifolds. Surveys in Differential Geometry, VI. International Press, Boston (1999)

  21. Maciá Ó.: A nearly quaternionic structure on SU(3). J. Geom. Phys. 60, 791–798 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marchiafava S., Romani G.: Sui fibrati con struttura quaternionale generalizzata. Ann. Mat. Pura Appl. (4) 107, 131–157 (1975)

    Article  MathSciNet  Google Scholar 

  23. Martín-Cabrera F., Swann A.F.: The intrinsic torsion of almost quaternion-Hermitian manifolds. Ann. Inst. Fourier (Grenoble) 58(5), 1455–1497 (2008)

    Article  MathSciNet  Google Scholar 

  24. Miquel V.: Some examples of Riemannian almost-product manifolds. Pacific J. Math. 111(1), 163–178 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Naveira A.M.: A classification of Riemannian almost-product manifolds. Rend. Mat. (7) 3(3), 577–592 (1983)

    MathSciNet  MATH  Google Scholar 

  26. Puhle, C.: Riemannian manifolds with structure group PSU. to appear in J. London Math. Soc.

  27. Salamon S.M.: Quaternionic Kähler manifolds. Invent Math. 67, 143–171 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salamon S.M.: Differential geometry of quaternionic manifolds. Ann. Sci. École Norm. Sup. (4) 19(1), 31–55 (1986)

    MathSciNet  MATH  Google Scholar 

  29. Salamon, S.M.: Almost parallel structures. Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, 2000), 162–181, Contemp. Math., 288, Amer. Math. Soc., Providence, RI (2001)

  30. Swann A.F.: Aspects symplectiques de la géométrie quaternionique. C. R. Acad. Sci. Paris Sér. I Math. 308(7), 225–228 (1989)

    MathSciNet  MATH  Google Scholar 

  31. Swann A.F.: Hyper-Kähler and quaternionic Kähler geometry. Math. Ann. 289(3), 421–450 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Thomas E.: Complex structures on real vector bundles. Amer. J. Math. 89, 887–908 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  33. Thomas E.: Vector fields on low dimensional manifolds. Math. Z. 103, 85–93 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  34. Witt F.: Special metrics and triality. Adv. Math. 219(6), 1972–2005 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Simon G. Chiossi

  2. Departamento de Geometria y Topologia, Universidad de Valencia, Av. Vicent Andrés Estellés 1, 46100, Burjassot, Valencia, Spain

    Óscar Maciá

Authors
  1. Simon G. Chiossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Óscar Maciá
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simon G. Chiossi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chiossi, S.G., Maciá, Ó. SO(3)-Structures on 8-manifolds. Ann Glob Anal Geom 43, 1–18 (2013). https://doi.org/10.1007/s10455-012-9329-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-012-9329-x

Keywords

Mathematics Subject Classification

Search

Navigation