Introduction to Möbius Differential Geometry

The book is an introduction to the geometry of submanifolds of the conformal n-sphere. A Möbius transformation is a conformal (i.e., angle preserving) transformation of the sphere. The sphere can be considered as a homogeneous space of the group of Möbius transforms, hence it is a model of Möbius geometry from F. Klein's point of view. There are several other models of Möbius geometry (projective model, quaternionic model and Clifford algebra model). Classically, a conformal structure is represented by a Riemannian metric (modulo a multiplication by a positive function). The change of Riemannian properties under conformal change is described and the Weyl and Schouten tensors are introduced. Conformal flatness is discussed in details. The projective model and related objects (congruencies, etc.) are introduced in the first chapter. The Cartan method of moving frames is often used for computations. As an application of projective model constructions, conformally flat hypersurfaces, isothermic and Willmore surfaces are discussed. A quaternionic model introduced in the next chapter is used for a study of isothermic surfaces in the four-dimensional case, while in higher dimensions, the Clifford algebra model is used. At the end of the book, the reader can find a discussion of triply orthogonal systems, their Ribaucour transformations and isothermic surfaces of arbitrary codimension. The book is a well-written survey of classical results from a new point of view and a nice textbook for a study of the subject.

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