Lie Sphere Transformations

  • Thomas E. Cecil
Part of the Universitext book series (UTX)


In this chapter, we study the sphere geometries of Lie, Moebius and Laguerre from the point of view of Klein’s Erlangen Program. In each case, we determine the group of transformations which preserve the fundamental geometric properties of the space. All of these groups are quotient groups or subgroups of some orthogonal group determined by an indefinite scalar product on a real vector space. As a result, the theorem of Cartan and Dieudonné, proven in Section 2.2, implies that each of these groups is generated by inversions. In Section 2.3, we give a geometric description of Moebius inversions. This is followed by a treatment of Laguerre geometry in Section 2.4. Finally, in Section 2.5, we show that the Lie sphere group is generated by the union of the groups of Moebius and Laguerre. There we also describe the place of Euclidean, spherical and hyperbolic metric geometries within the context of these more general geometries.


Lightlike Vector Point Sphere Oriented Sphere Moebius Transformation Lightlike Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Thomas E. Cecil
    • 1
  1. 1.Department of MathematicsCollege of the Holy CrossWorcesterUSA

Personalised recommendations