Mathematische Zeitschrift

, Volume 259, Issue 3, pp 617–629 | Cite as

A sufficient condition for nonrigidity of Carnot groups

Article

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.

Mathematics Subject Classification (2000)

22E25 22E60 53D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Corwin L., Greenleaf F.P. (1990). Representations of Nilpotent Lie Groups and Their Applications. Cambridge University Press, Cambridge MATHGoogle Scholar
  2. 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 MATHCrossRefGoogle Scholar
  3. 3.
    De Mari F., Pedroni M. (1999). Toda flows and real Hessenberg manifolds. J. Geom. Anal. 9(4): 607–625 MATHMathSciNetGoogle Scholar
  4. 4.
    Ottazzi A. (2005). Multicontact vector fields on Hessenberg manifolds. J. Lie Theory 15: 357–377 MATHMathSciNetGoogle Scholar
  5. 5.
    Reimann H.M. (2001). Rigidity of H-type groups. Math. Z. 237(4): 697–725 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Reimann, H.M., Ricci, F.: The complexified Heisenberg group. In: Proceedings on Analysis and Geometry (Russian) Novosibirsk Akademgorodok, pp. 465–480 (1999)Google Scholar
  7. 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 MATHCrossRefMathSciNetGoogle Scholar
  8. 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. 9.
    Warhurst B. (2005). Jet spaces as nonrigid Carnot groups. J. Lie Theory 15: 341–356 MATHMathSciNetGoogle Scholar
  10. 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. 413–494Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BernBernSwitzerland
  2. 2.DIMAUniversità di GenovaGenova (1)Italy

Personalised recommendations