Introduction to Möbius Differential Geometry (London Mathematical Society Lecture Note Series)

Hertrich-Jeromin, Udo

Cambridge University Press（2003/08発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 428 p.
• 言語 ENG
• 商品コード 9780521535694
• DDC分類 516.36

Full Description

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.

Contents

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.