This book introduces the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Möbius geometry are presented: the classical projective model, the quaternionic approach, and an approach that uses the Clifford algebra of the space of homogeneous coordinates of the classical model; the use of 2-by-2 matrices in this context is elaborated. For each model in turn applications are discussed. Topics comprise conformally flat hypersurfaces, isothermic surfaces and their transformation theory, Willmore surfaces, orthogonal systems and the Ribaucour transformation, as well as analogous discrete theories for isothermic surfaces and orthogonal systems. Certain relations with curved flats, a particular type of integrable system, are revealed. Thus this book will serve both as an introduction to newcomers (with background in Riemannian geometry and elementary differential geometry) and as a reference work for researchers.

'One of the most attractive features of the book is the detailed discussion of discrete analogues of isothermic surfaces and orthogonal systems which were intensively studied in the last decade … The reviewed monograph will be of great interest for researchers specializing in differential geometry, geometric theory of integrable systems and other related fields.'

Source: Zentralblatt MATH

'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.'

Source: EMS Newsletter

'This book is a work of scholarship, communicating the author's enthusiasm for Möbius geometry very clearly. The book will serve as an introduction to Möbius geometry to newcomers, and as a very useful reference for research workers in the field.'

Tom Willmore - University of Durham