Geometry

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By Thomas E. Cecil

This exposition presents the state-of-the artwork at the differential geometry of hypersurfaces in genuine, complicated, and quaternionic area varieties. unique emphasis is put on isoparametric and Dupin hypersurfaces in actual house kinds in addition to Hopf hypersurfaces in advanced house varieties. The publication is on the market to a reader who has accomplished a one-year graduate path in differential geometry. The textual content, together with open difficulties and an in depth record of references, is a wonderful source for researchers during this area.

Geometry of Hypersurfaces starts off with the fundamental concept of submanifolds in genuine house types. subject matters contain form operators, primary curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. the focal point then turns to the idea of isoparametric hypersurfaces in spheres. vital examples and category effects are given, together with the development of isoparametric hypersurfaces in line with representations of Clifford algebras. An in-depth therapy of Dupin hypersurfaces follows with effects which are proved within the context of Lie sphere geometry in addition to those who are got utilizing ordinary equipment of submanifold conception. subsequent comes an intensive remedy of the idea of genuine hypersurfaces in advanced house forms.  A valuable concentration is a whole facts of the type of Hopf hypersurfaces with consistent imperative curvatures as a result of Kimura and Berndt. The e-book concludes with the elemental concept of genuine hypersurfaces in quaternionic area kinds, together with statements of the most important type effects and instructions for extra research.

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