Introduction to Möbius Differential Geometry

This book introduces the reader to the geometry of surfaces and submanifolds in the conformal n-sphere.

Udo Hertrich-Jeromin (Author)

9780521535694, Cambridge University Press

Paperback, published 14 August 2003

428 pages, 36 b/w illus.
22.9 x 15.2 x 2.4 cm, 0.582 kg

'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

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.

Introduction
0. Preliminaries: the Riemannian point of view
1. The projective model
2. Application: conformally flat hypersurfaces
3. Application: isothermic and Willmore surfaces
4. A quaternionic model
5. Application: smooth and discrete isothermic surfaces
6. A Clifford algebra model
7. A Clifford algebra model
Vahlen matrices
8. Applications: orthogonal systems, isothermic surfaces
Conclusion.

Subject Areas: Differential & Riemannian geometry [PBMP]