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.
