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Geometry of Hypersurfaces

Springer Monographs in Mathematics

Format: Paperback / softback
Publisher: Springer-Verlag New York Inc., New York, United States
Published: 23rd Aug 2016
Dimensions: w 156mm h 234mm d 31mm
Weight: 844g
ISBN-10: 1493945076
ISBN-13: 9781493945078
Barcode No: 9781493945078
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Synopsis
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.

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"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)