Abstract
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.
Highlights
It is worthwhile to mention that isothermic immersions are invariant with respect to conformal transformations of the ambient space and can be naturally described in terms of conformal geometry (
In this paper we develop an approach based on using Clifford algebras and Spin groups [20,21]
Isothermic surfaces are characterized as surfaces immersed in E3 with curvature lines admitting conformal parameterization
Summary
Isothermic surfaces have a very long history. Lamé in studies on stationary heat flows (described by the Laplace equation), in the broader context of triply ortogonal systems of coordinates [1]. Transformations of isothemic surfaces, studied by Darboux and Bianchi [4,5], strongly suggested that the related system of nonlinear partial differential equations (see (2) below) is integrable in the sense of the soliton theory [6] and, such modern formulation of this problem was found [7], which started new developments in this field [8,9,10,11,12]. We re-derive the construction of “multisoliton” surfaces by iterated Darboux transformation.
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