Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 A Bundle Approach to Conformal Surfaces in Space-Forms
- 2 The Mean Curvature Sphere Congruence
- 3 Surfaces under Change of Flat Metric Connection
- 4 Willmore Surfaces
- 5 The Euler–Lagrange ConstrainedWillmore Surface Equation
- 6 Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces
- 7 Constrained Willmore Surfaces with a Conserved Quantity
- 8 Constrained Willmore Surfaces and the Isothermic Surface Condition
- 9 The Special Case of Surfaces in 4-Space
- Appendix A Hopf Differential and Umbilics
- Appendix B Twisted vs. Untwisted Bäcklund Transformation Parameters
- References
- Index
1 - A Bundle Approach to Conformal Surfaces in Space-Forms
Published online by Cambridge University Press: 13 May 2021
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 A Bundle Approach to Conformal Surfaces in Space-Forms
- 2 The Mean Curvature Sphere Congruence
- 3 Surfaces under Change of Flat Metric Connection
- 4 Willmore Surfaces
- 5 The Euler–Lagrange ConstrainedWillmore Surface Equation
- 6 Transformations of Generalized Harmonic Bundles and Constrained Willmore Surfaces
- 7 Constrained Willmore Surfaces with a Conserved Quantity
- 8 Constrained Willmore Surfaces and the Isothermic Surface Condition
- 9 The Special Case of Surfaces in 4-Space
- Appendix A Hopf Differential and Umbilics
- Appendix B Twisted vs. Untwisted Bäcklund Transformation Parameters
- References
- Index
Summary
Our study is one of the geometrical aspects that are invariant under Möbius transformations. We present a Möbius description of space-forms in the projectivized light-cone. Such description is based on Darboux's model of the conformal n-sphere on the projective space of the light-cone in (n+1,1)-space, which, in particular, yields a conformal description of Euclidean n-spaces and hyperbolic n-spaces as submanifolds of the projectivized light-cone. With this, we approach a surface conformally immersed in a space-form as a null line subbundle of the Lorentzian trivial bundle over the surface.
Keywords
- Type
- Chapter
- Information
- Constrained Willmore SurfacesSymmetries of a Möbius Invariant Integrable System, pp. 16 - 30Publisher: Cambridge University PressPrint publication year: 2021