Annals of Global Analysis and Geometry

, Volume 53, Issue 3, pp 331–375 | Cite as

On the equivalence of quaternionic contact structures

  • Ivan Minchev
  • Jan Slovák


Following the Cartan’s original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components.


Quaternionic contact Equivalence problem Cartan connection Involution 

Mathematics Subject Classification

58G30 53C17 



I.M. is partially supported by Contract DFNI I02/4/12.12.2014 and Contract 80-10-33/2017 with the Sofia University “St.Kl.Ohridski”. I.M. is also supported by a SoMoPro II Fellowship which is cofunded by the European Commission (this article reflects only the author’s views, and the EU is not liable for any use that may be made of the information contained therein) from “People” specific program (Marie Curie Actions) within the EU Seventh Framework Program on the basis of the Grant Agreement REA No. 291782. It is further cofinanced by the South-Moravian Region. J.S. is supported by the Grant P201/12/G028 of the Grant Agency of the Czech Republic.


  1. 1.
    Armstrong, S.: Non-regular \(|2|-\)graded geometries I: general theory. arXiv:0902.1133 (2009), 23 pp
  2. 2.
    Alt, J.: Weyl connections and the local sphere theorem for quaternionic contact structures. Ann. Glob. Anal. Geom. 39, 165–186 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque 265 (2000)Google Scholar
  4. 4.
    Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582, 143–172 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Čap, A., Slovák, J.: Parabolic Geometries. I. Background and General Theory, Mathematical Surveys and Monographs. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  6. 6.
    Chern, S.-S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974); Erratum Acta Math. 150, 297 (1983)Google Scholar
  7. 7.
    Duchemin, D.: Quaternionic contact structures in dimension 7. Ann. Inst. Fourier Grenoble 56(4), 851–885 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ivanov, S., Vassilev, D.: Conformal quaternionic contact curvature and the local sphere theorem. J. Math. Pures Appl. 93, 277–307 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ivanov, S., Minchev, I., Vassilev, D.: Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem. J. Eur. Math. Soc. 12, 1041–1067 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ivanov, S., Minchev, I., Vassilev, D.: Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem. Mem. Amer. Math. Soc. 231, 1086 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ivanov, S., Minchev, I., Vassilev, D.: The optimal constant in the \(L^2\) Folland-Stein inequality on the quaternionic Heisenberg group. Ann. Sc. Norm. Super Pisa Cl. Sci. (5) XI, 635–652 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ivanov, S., Minchev, I., Vassilev, D.: Quaternionic contact hypersurfaces in hyper-Kähler manifolds. Ann. Mat. Pura Appl. 196(4), 245–267 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22, 413–494 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsUniversity of SofiaSofiaBulgaria
  2. 2.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations