Universitext
cover

Lie Sphere Geometry

With Applications to Submanifolds

Authors: Cecil, Thomas E

  • Entirely new section on isoparametric hypersurfaces in spheres, including a thorough presentation and proofs from recent papers that discuss the main properties of isoparametric hypersurfaces with four principal curvatures
  • Revised sections on taut submanifolds, Lie frames, compact proper dupin submanifolds and reducible dupin submanifolds with up-to-date results
  • New sections on dupin hypersurfaces with three and four principal curvatures
  • Includes a more systematic treatment of frame reductions
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  • ISBN 978-0-387-74656-2
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Softcover $84.99
price for USA in USD
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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

About the authors

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.

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


Table of contents (5 chapters)

Table of contents (5 chapters)
  • Introduction

    Pages 1-7

  • Lie Sphere Geometry

    Pages 9-23

  • Lie Sphere Transformations

    Pages 25-49

  • Legendre Submanifolds

    Pages 51-123

  • Dupin Submanifolds

    Pages 125-190

Buy this book

eBook $64.99
price for USA in USD (gross)
  • ISBN 978-0-387-74656-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $84.99
price for USA in USD
  • ISBN 978-0-387-74655-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Lie Sphere Geometry
Book Subtitle
With Applications to Submanifolds
Authors
Series Title
Universitext
Copyright
2008
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-0-387-74656-2
DOI
10.1007/978-0-387-74656-2
Softcover ISBN
978-0-387-74655-5
Series ISSN
0172-5939
Edition Number
2
Number of Pages
XII, 208
Topics