Lie Sphere Geometry

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   • Thomas E. Cecil

   Pages 1-7

3. Lie Sphere Geometry

   • Thomas E. Cecil

   Pages 8-28

4. Lie Sphere Transformations

   • Thomas E. Cecil

   Pages 29-64

5. Legendre Submanifolds

   • Thomas E. Cecil

   Pages 65-128

6. Dupin Submanifolds

   • Thomas E. Cecil

   Pages 129-190

7. Back Matter

   Pages 191-209

   PDF

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

• Department of Mathematics, College of the Holy Cross, Worcester, USA

  Thomas E. Cecil

Policies and ethics