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Geometry of Hypersurfaces

  • Thomas E. Cecil
  • Patrick J. Ryan

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Thomas E. Cecil, Patrick J. Ryan
    Pages 1-7
  3. Thomas E. Cecil, Patrick J. Ryan
    Pages 9-83
  4. Thomas E. Cecil, Patrick J. Ryan
    Pages 85-184
  5. Thomas E. Cecil, Patrick J. Ryan
    Pages 185-231
  6. Thomas E. Cecil, Patrick J. Ryan
    Pages 233-342
  7. Thomas E. Cecil, Patrick J. Ryan
    Pages 343-386
  8. Thomas E. Cecil, Patrick J. Ryan
    Pages 387-420
  9. Thomas E. Cecil, Patrick J. Ryan
    Pages 421-531
  10. Thomas E. Cecil, Patrick J. Ryan
    Pages 533-551
  11. Back Matter
    Pages 553-596

About this book

Introduction

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.

Keywords

Dupin hypersurfaces Hopf hypersurfaces Lie sphere geometry differential geometry submanifolds geometry of hypersurfaces isoparametric hypersurfaces in spheres

Authors and affiliations

  • Thomas E. Cecil
    • 1
  • Patrick J. Ryan
    • 2
  1. 1.Department of Mathematics & Computer SciCollege of the Holy CrossWorcesterUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-3246-7
  • Copyright Information Thomas E. Cecil and Patrick J. Ryan 2015
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-3245-0
  • Online ISBN 978-1-4939-3246-7
  • Series Print ISSN 1439-7382
  • Series Online ISSN 2196-9922
  • Buy this book on publisher's site
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