Lie Sphere Geometry

With Applications to Submanifolds

  • Thomas E. Cecil

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Thomas E. Cecil
    Pages 1-7
  3. Thomas E. Cecil
    Pages 8-28
  4. Thomas E. Cecil
    Pages 29-64
  5. Thomas E. Cecil
    Pages 65-128
  6. Thomas E. Cecil
    Pages 129-190
  7. Back Matter
    Pages 191-209

About this book


Lie Sphere Geometry provides a modern treatment of Lie's geometry of spheres, its recent applications and the study of Euclidean space. This book begins with Lie's construction of the space of spheres, including the fundamental notions of oriented contact, parabolic pencils of spheres and Lie sphere transformation. The link with Euclidean submanifold theory is established via the Legendre map. This provides a powerful framework for the study of submanifolds, especially those characterized by restrictions on their curvature spheres. Of particular interest are isoparametric, Dupin and taut submanifolds. These have recently been classified up to Lie sphere transformation in certain special cases through the introduction of natural Lie invariants. The author provides complete proofs of these classifications and indicates directions for further research and wider application of these methods.


Invariant Lie Natural character classification construction curvature form framework geometry manifold proof transformation

Authors and affiliations

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

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York 1992
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-97747-8
  • Online ISBN 978-1-4757-4096-7
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site