Geometriae Dedicata

, Volume 111, Issue 1, pp 65–86

Contact and Conformal Maps in

Parabolic Geometry. I
  • Michael Cowling
  • Filippo De Mari
  • Adam Korányi
  • Hans Martin Reimann


When \(n \geqslant 3,\) the action of the conformal group O(1, n+1) on \({\mathbb R}^n \cup\{\infty\}\) may be characterized in simple differential geometric terms, even locally: a theorem of Liouville states that a C4 map between domains \({\cal U}\) and \({\cal V}\) in \({\mathbb R}^n\) whose differential is a (variable) multiple of a (variable) isometry at each point of \({\cal U}\) is the restriction to \({\cal U}\) of a transformation xg·x, for some g in O(1,n+1). In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G/P, where P is a minimal parabolic subgroup.

Mathematics Subject Classifications (2000)

53C30 53A30 57S20 


semisimple Lie group contact map conformal map 


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Copyright information

© Springer 2005

Authors and Affiliations

  • Michael Cowling
    • 1
  • Filippo De Mari
    • 2
  • Adam Korányi
    • 3
  • Hans Martin Reimann
    • 4
  1. 1.School of MathematicsUniversity of New South WalesUNSW SydneyAustralia
  2. 2.DIPEMUniversità di GenovaGenovaItaly
  3. 3.Department of Mathematics and Computer ScienceLehman CollegeBronxU.S.A
  4. 4.Mathematisches InstitutUniversität BernBernSwitzerland

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