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

Format:

Book

Author/Creator:

Cecil, T. E. (Thomas E.), 1945-

Series:

Universitext

Language:

English

Subjects (All):

Geometry, Differential.

Submanifolds.

Physical Description:

xii, 208 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer, 2008.

Summary:

This book provides a clear and comprehensive modern treatment of Lie sphere geometry and its applications to the study of Euclidean submanifolds. It begins with the construction of the space of spheres, including the fundamental notions of oriented contact, parabolic pencils of spheres, and Lie sphere transformations. The link with Euclidean submanifold theory is established via the Legendre map, which provides a powerful framework for the study of submanifolds, especially those characterized by restrictions on their curvature spheres.

This new edition contains revised sections on taut submanifolds, compact proper Dupin submanifolds, reducible Dupin submanifolds, and the cyclides of Dupin. Completely new material on isoparametric hypersurfaccs in spheres and Dupin hypersurfaces with three and four principal curvatures is also included. The author surveys the known results in these fields and indicates directions for further research and wider application of the methods of Lie sphere geometry.

Further key features of Lie Sphere Geometry 2/e: Provides the reader with all the necessary background to reach the frontiers of research in this area, Fills a gap in the literature; no other thorough examination of Lie sphere geometry and its applications to submanifold theory, Complete treatment of the cyclides of Dupin, including 11 computer-generated illustrations, Rigorous exposition driven by motivation and ample examples.

Contents:

2 Lie Sphere Geometry 9

2.2 Mobius Geometry of Unoriented Spheres 11

2.3 Lie Geometry of Oriented Spheres 14

2.4 Geometry of Hyperspheres in S" and H" 16

2.5 Oriented Contact and Parabolic Pencils of Spheres 19

3 Lie Sphere Transformations 25

3.1 The Fundamental Theorem 25

3.2 Generation of the Lie Sphere Group by Inversions 30

3.3 Geometric Description of Inversions 34

3.4 Laguerre Geometry 37

3.5 Subgeometries of Lie Sphere Geometry 46

4 Legendre Submanifolds 51

4.1 Contact Structure on [Lambda superscript 2n-1] 51

4.2 Definition of Legendre Submanifolds 56

4.3 The Legendre Map 60

4.4 Curvature Spheres and Parallel Submanifolds 64

4.5 Lie Curvatures and Isoparametric Hypersurfaces 72

4.6 Lie Invariance of Tautness 82

4.7 Isoparametric Hypersurfaces of FKM-type 95

4.8 Compact Proper Dupin Submanifolds 112

5 Dupin Submanifolds 125

5.1 Local Constructions 125

5.2 Reducible Dupin Submanifolds 127

5.3 Lie Sphere Geometric Criterion for Reducibility 141

5.4 Cyclides of Dupin 148

5.5 Lie Frames 159

5.6 Covariant Differentiation 165

5.7 Dupin Hypersurfaces in 4-Space 168.

Notes:

Includes bibliographical references (pages [191]-199) and index.

ISBN:

0387746552

9780387746555

OCLC:

173498965