Introduction to Mobius Differential Geometry

Erscheinungsjahr 2003
ISBN: 978-0-521-53569-4
Verlag: Cambridge University Press

Buch, Englisch, Band 300, 426 Seiten, Format (B × H): 152 mm x 229 mm, Gewicht: 691 g

Reihe: London Mathematical Society Lecture Note Series

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.

Autoren/Hrsg.

Hertrich-Jeromin, Udo

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie

Inhaltsverzeichnis

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.