Introduction to Möbius Differential Geometry

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.

• Different approaches ('models') are given for each geometric problem presented so the reader can compare: a) classical approach, b) quaternionic approach, c) Clifford algebra approach
• Material from the classical literature is compiled into the text and many historical references are given; the reader is led to topics/questions of current research interest
• Certain relations between geometry and integrable systems theory are discussed as well as topics in discrete differential geometry