Conformal Geometry of Surfaces in S4 and Quaternions [Paperback]

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 B?cklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.- B?cklund transforms of Willmore surfaces.- Willmore surfaces in S3.- Spherical Willmore surfaces in HP1.- Darboux transforms.- Appendix: The bundle L. Holomorphicity and the Ejiri theorem.Springer Book Archives