Conformal Geometry of Surfaces in S4 and Quaternions

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

List of contents

Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.- Bäcklund transforms of Willmore surfaces.- Willmore surfaces in S3.- Spherical Willmore surfaces in HP1.- Darboux transforms.- Appendix: The bundle L. Holomorphicity and the Ejiri theorem.

Summary

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

Product details

• Authors: Francis Burstall, Francis E Burstall, Francis E. Burstall, Dir Ferus, Dirk Ferus, Katrin Leschke, Katrin et Leschke, Franz Pedit, Ulrich Pinkall
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2002
• EAN: 9783540430087
• ISBN: 978-3-540-43008-7
• No. of pages: 89
• Weight: 170 g
• Series: Lecture Notes in Mathematics; Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; B, Mathematics and Statistics, Differential Geometry, curvature, twistor projection, darboux transforms, Wilmore functional