Lie Sphere Geometry: With Applications to Submanifolds

Thomas Cecil9780387746555, 0387746552, 0387746560

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 hypersurfaces 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.

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Table of contents :
Preface to the First Edition……Page 5
Preface to the Second Edition……Page 7
Contents……Page 9
Introduction……Page 11
Lie Sphere Geometry……Page 18
Lie Sphere Transformations……Page 33
Legendre Submanifolds……Page 58
Dupin Submanifolds……Page 131
References……Page 197
Index……Page 206