Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems

Abstract

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.

77 (A4) pages, 3 (small) PostScript figures. v3: Final final version: some typos corrected, references and exposition updated

References in corpus (7)

• Isothermic surfaces in $\E^3$ as soliton surfaces
• The submanifold geometries associated to Grassmannian systems
• A discrete version of the Darboux transform for isothermic surfaces
• Surfaces in Lie sphere geometry and the stationary Davey-Stewartson hierarchy
• Conformal Geometry of Surfaces in the 4-Sphere and Quaternions
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Cited by in corpus (8)

• The Ribaucour transformation in Lie sphere geometry
• Adjoint Transform of Willmore Surfaces in $n$-sphere
• Dressing preserving the fundamental group
• Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics
• Generalized polar transforms of spacelike isothermic surfaces
• Constructing solutions to the Björling problem for isothermic surfaces by structure preserving discretization
• Group interpretation of the spectral parameter. The case of isothermic surfaces
• Isothermic and S-Willmore Surfaces as Solutions to Blaschke's Problem