Skip to main content
Springer Nature Link
Log in

A sufficient condition for nonrigidity of Carnot groups

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this article we consider contact mappings on Carnot groups. Namely, we are interested in those mappings whose differential preserves the horizontal space, defined by the first stratum of the natural stratification of the Lie algebra of a Carnot group. We give a sufficient condition for a Carnot group G to admit an infinite dimensional space of contact mappings, that is, for G to be nonrigid. A generalization of Kirillov’s Lemma is also given. Moreover, we construct a new example of nonrigid Carnot group.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Corwin L., Greenleaf F.P. (1990). Representations of Nilpotent Lie Groups and Their Applications. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Cowling M., De Mari F., Korányi A., Reimann H.M. (2005). Contact and conformal mappings in parabolic geometry. I. Geom. Dedicata 111: 65–86

    Article  MATH  Google Scholar 

  3. De Mari F., Pedroni M. (1999). Toda flows and real Hessenberg manifolds. J. Geom. Anal. 9(4): 607–625

    MATH  MathSciNet  Google Scholar 

  4. Ottazzi A. (2005). Multicontact vector fields on Hessenberg manifolds. J. Lie Theory 15: 357–377

    MATH  MathSciNet  Google Scholar 

  5. Reimann H.M. (2001). Rigidity of H-type groups. Math. Z. 237(4): 697–725

    Article  MATH  MathSciNet  Google Scholar 

  6. Reimann, H.M., Ricci, F.: The complexified Heisenberg group. In: Proceedings on Analysis and Geometry (Russian) Novosibirsk Akademgorodok, pp. 465–480 (1999)

  7. Rigot S. (2004). Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric. Math. Z. 248: 827–848

    Article  MATH  MathSciNet  Google Scholar 

  8. Saunders D.J. (1989). The Geometry of Jet Bundles, vol. 142. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge

    Google Scholar 

  9. Warhurst B. (2005). Jet spaces as nonrigid Carnot groups. J. Lie Theory 15: 341–356

    MATH  MathSciNet  Google Scholar 

  10. Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. In: Progress in differential geometry. Adv. Stud. Pure Math. 22, Math. Soc. Japan, Tokyo 1993, pp. 413–494

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland

    Alessandro Ottazzi

  2. DIMA, Università di Genova, Via Dodecaneso 35, 16146, Genova (1), Italy

    Alessandro Ottazzi

Authors
  1. Alessandro Ottazzi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Alessandro Ottazzi.

Additional information

This research was partly supported by the Swiss National Science Foundation. The author would like to thank H. M. Reimann for the helpful advices and the constant support.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ottazzi, A. A sufficient condition for nonrigidity of Carnot groups. Math. Z. 259, 617–629 (2008). https://doi.org/10.1007/s00209-007-0240-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-007-0240-2

Mathematics Subject Classification (2000)