Iterated Darboux Transformation for Isothermic Surfaces in Terms of Clifford Numbers

Isothermic surfaces are defined as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly showing Bianchi's permutability theorem. In algebraic calculations an important role is played by the main anti-automorphism (reversion) of the Clifford algebra C(4,1) and the spinorial norm in the corresponding Spin group.

Automated summary

Isothermic surfaces are immersions that can be conformally parameterized with curvature lines. The paper discusses the reconstruction of the Darboux transformation using Clifford numbers, and presents a symmetric formula for the two-fold Darboux transformation demonstrating Bianchi's permutability theorem. The main anti-automorphism of the Clifford algebra C(4,1) and the spinorial norm in the corresponding Spin group play important roles in algebraic calculations.