Geometry of Hypersurfaces

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.

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?

Auteur
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.

Contenu
Preface.- 1. Introduction.- 2. Submanifolds of Real Space Forms.- 3. Isoparametric Hypersurfaces.- 4. Submanifolds in Lie Sphere Geometry.- 5. Dupin Hypersurfaces.- 6. Real Hypersurfaces in Complex Space Forms.- 7. Complex Submanifolds of CP n and CH n .- 8. Hopf Hypersurfaces.- 9. Hypersurfaces in Quaternionic Space Forms.- Appendix A. Summary of Notation.- References.- Index.