lie sphere geometry: with applications to submanifolds/ Thomas E.Cecil

Chapter 1 — Lie Sphere Geometry 8
1.1 Preliminaries ®
1.2 Moebius Geometry of Unoriented Spheres 11
1.3 Lie Geometry of Oriented Spheres 15
1.4 Geometry of Hyperspheres in S and// 19
1.5 Oriented Contact and Parabolic Pencils of Spheres 22
Chapter 2 - Lie Sphere Transformations 29
2.1 The Fundamental Theorem 29
2.2 Generation of the Lie Sphere Group by Inversions 36
2.3 Geometric Description of Inversions 42
2.4 Laguerre Geometry 47
2.5 Sub Geometries of Lie Sphere Geometry 59
Chapters - Legendre Submanifolds 65
3.1 Contact Structure 65
3.2 Definition of Legendre Submanifolds 73
3.3 The Legendre Map 7,8
3.4 Curvature Spheres and Parallel Submanifolds 84
3.5 Lie Curvatures of Legendre Submanifolds 95
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3.6 Taut Submanifolds 108
3.7 Compact Proper Dupin Submanifolds 113"
Chapter 4 — Dupin Submanifolds 129
4.1 Local Constructions 129
4.2 Reducible Dupin Submanifolds 130
4.3 Cyclides of Dupin 150
4.4 Principal Lie Frames 159
4.5 Half-Invariant Differentiation 166
4.6 Dupin Hypersurfaces in 4-Space 171