Skip to main content

Contact projective structures and chains


Contact projective structures have been thoroughly studied by D. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox’s construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman’s construction of a conformal structure associated to a CR structure. We show that, on the level of Cartan connections, this Fefferman-type construction is compatible with normality if and only if the initial structure has vanishing contact torsion. This leads to a geometric description of the paths that have to be added to the contact geodesics of a contact projective structure in order to obtain the subordinate projective structure. They are exactly the chains associated to the contact projective structure, which are analogs of the Chern–Moser chains in CR geometry. Finally, we analyze the consequences for the geometry of chains and prove that a chain–preserving contactomorphism must be a morphism of contact projective structures.

This is a preview of subscription content, access via your institution.


  1. Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582, 143–172 (2005). MR 2139714 (2006h:32017)

    Google Scholar 

  2. Čap, A.: Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo Suppl. Series 2. 79, 11–37 (2006). MR 2287124 (2008a:53049)

  3. Čap, A., Gover, A.R.: Tractor bundles for irreducible parabolic geometries. Global analysis and harmonic analysis (Marseille-Luminy, 1999), Sémin. Congr., vol. 4, Soc. Math. France, Paris, 129–154 (2000). MR 1822358 (2002b:53033)

  4. Čap, A., Gover, A.R.: Tractor calculi for parabolic geometries. Trans. Am. Math. Soc. 354(4), 1511–1548 (2002) (electronic). MR 1873017 (2003j:53033)

    Google Scholar 

  5. Čap, A., Gover, A.R.: CR tractors an the Fefferman construction. Indiana Univ. Math. J. 57(5), 2519–2570 (2008). MR 2463976

    Google Scholar 

  6. Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29(3), 453–505 (2000). MR 1795487 (2002f:53036)

    Google Scholar 

  7. Čap, A., Slovák, J., Žádník, V.: On distinguished curves in parabolic geometries. Transform. Groups 9(2), 143–166 (2004). MR 2056534 (2005a:53042)

    Google Scholar 

  8. Čap, A., Žádník, V.: On the geometry of chains. J. Differ. Geom. 82(1), 1–33 (2009). MR 2504769

    Google Scholar 

  9. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). MR 0425155 (54 #13112)

    Google Scholar 

  10. Fox, D.J.F.: Contact projective structures. Indiana Univ. Math. J. 54(6), 1547–1598 (2005). MR 2189678 (2007b:53163)

  11. Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74(2), 329–387 (1961). MR 0142696 (26 #265)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Andreas Čap.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Čap, A., Žádník, V. Contact projective structures and chains. Geom Dedicata 146, 67–83 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Projective structure
  • Contact projective structure
  • Path geometry
  • Fefferman construction
  • Chains
  • Cartan connection
  • Parabolic geometry

Mathematics Subject Classification (2000)

  • 53A20
  • 53B10
  • 53B15
  • 53C15
  • 53D10