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Springer Monographs in Mathematics

Geometry of Hypersurfaces

Authors: Cecil, Thomas E, Ryan, Patrick J.

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  • ​​​Presents thorough treatment of hypersurfaces in real, complex, and quaternionic space forms with connections to symmetric spaces, homogeneous spaces, and Riemannian geometry
  • Treats Dupin hypersurfaces using both standard and Lie sphere geometric techniques
  • Discusses the comprehensive treatment of the theory of isoparametric hypersurfaces due to Cartan and Münzner that are necessary for understanding the subject​
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eBook $109.00
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  • ISBN 978-1-4939-3246-7
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Hardcover $149.99
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  • ISBN 978-1-4939-3245-0
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  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
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  • ISBN 978-1-4939-4507-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area.

Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hypersurfaces follows with results that are proved in the context of Lie sphere geometry as well as those that are obtained using standard methods of submanifold theory. Next comes a thorough treatment of the theory of real hypersurfaces in complex space forms.  A central focus is a complete proof of the classification of Hopf hypersurfaces with constant principal curvatures due to Kimura and Berndt. The book concludes with the basic theory of real hypersurfaces in quaternionic space forms, including statements of the major classification results and directions for further research.

About the authors

Thomas E. Cecil is professor of mathematics at the College of Holy Cross in Worcester, MA, USA. His primary research interests are in differential geometry, in particular, submanifolds.

Patrick J. Ryan is Emeritus professor of mathematical sciences at McMaster University in Hamilton, Ontario, Canada. His primary research interests are in Geometry, in particular, the characterization and classification of hypersurfaces in real and complex space forms.

Reviews

“This 600-page book is the result of the authors’ efforts to provide a detailed presentation of the present day differential geometry of hypersurfaces in real, complex, and quaternionic space forms. … A summary of the frequently used notations and an index of notions are included. The book is an essential contribution to the progress of the theory of hypersurfaces.” (Radu Miron, zbMATH 1331.53001, 2016)


Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook $109.00
price for Mexico (gross)
  • ISBN 978-1-4939-3246-7
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.99
price for Mexico
  • ISBN 978-1-4939-3245-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for Mexico
  • ISBN 978-1-4939-4507-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Geometry of Hypersurfaces
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2015
Publisher
Springer-Verlag New York
Copyright Holder
Thomas E. Cecil and Patrick J. Ryan
eBook ISBN
978-1-4939-3246-7
DOI
10.1007/978-1-4939-3246-7
Hardcover ISBN
978-1-4939-3245-0
Softcover ISBN
978-1-4939-4507-8
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XI, 596
Number of Illustrations
23 b/w illustrations
Topics