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  • © 2008

Lie Sphere Geometry

With Applications to Submanifolds

Authors:

  • 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

Part of the book series: Universitext (UTX)

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eBook USD 64.99
Price excludes VAT (USA)
  • ISBN: 978-0-387-74656-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 84.99
Price excludes VAT (USA)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xii
  2. Introduction

    Pages 1-7
  3. Lie Sphere Geometry

    Pages 9-23
  4. Legendre Submanifolds

    Pages 51-123
  5. Dupin Submanifolds

    Pages 125-190
  6. Back Matter

    Pages 191-208

About this book

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.

Reviews from the first edition:

"The book under review sets out the basic material on Lie sphere geometry in modern notation, thus making it accessible to students and researchers in differential geometry.....This is a carefully written, thorough, and very readable book. There is an excellent bibliography that not only provides pointers to proofs that have been omitted, but gives appropriate references for the results presented. It should be useful to all geometers working in the theory of submanifolds."

- P.J. Ryan, MathSciNet

"The book under review is an excellent monograph about Lie sphere geometry and its recent applications to the study of submanifolds of Euclidean space.....The book is written in a very clear and precise style. It contains about a hundred references, many comments of and hints to the topical literature, and can be considered as a milestone in the recent development of a classical geometry, to which the author contributed essential results."

- R. Sulanke, Zentralblatt

Keywords

  • Dimension
  • Grad
  • curvature
  • differential geometry
  • manifold
  • projective geometry

Reviews

Reviews from the first edition:

"The book under review sets out the basic material on Lie sphere geometry in modern notation, thus making it accessible to students and researchers in differential geometry.....This is a carefully written, thorough, and very readable book. There is an excellent bibliography that not only provides pointers to proofs that have been omitted, but gives appropriate references for the results presented. It should be useful to all geometers working in the theory of submanifolds."

- P.J. Ryan, MathSciNet

"The book under review is an excellent monograph about Lie sphere geometry and its recent applications to the study of submanifolds of Euclidean space.....The book is written in a very clear and precise style. It contains about a hundred references, many comments of and hints to the topical literature, and can be considered as a milestone in the recent development of a classical geometry, to which the author contributed essential results."

- R. Sulanke, Zentralblatt

Authors and Affiliations

  • College of the Holy Cross, Worcester, USA

    Thomas E. Cecil

About the author

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.

Bibliographic Information

Buying options

eBook USD 64.99
Price excludes VAT (USA)
  • ISBN: 978-0-387-74656-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 84.99
Price excludes VAT (USA)