This book provides a modern treatment of Lie's geometry of spheres, its applications and the study of Euclidean space. It 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. This new edition contains revised sections on taut submanifolds, compact proper Dupin submanifolds, reducible Dupin submanifolds, Lie frames and frame reductions. Completely new material on isoparametric hyperspaces in spheres, Dupin hyperspaces with three and four principle curvatures is also included.

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 Includes supplementary material: sn.pub/extras

Autorentext

Professor Thomas E. Cecil is a professor of mathematics at Holy Cross University, where he has taught for almost thirty years. He has held visiting appointments at UC Berkeley, Brown University, and the University of Notre Dame. He has written several articles on Dupin submanifolds and hypersurfaces, and their connections to Lie sphere geometry, and co-edited two volumes on tight and taught submanifolds.

Inhalt
Lie Sphere Geometry.- Lie Sphere Transformations.- Legendre Submanifolds.- Dupin Submanifolds.