24 1. Classical Differential Geometry for i = 1, 2. Comparing this with the definition (1.56) of Ribaucour trans- formations we see that one of the two possibilities holds: (i) r1 = r2 = r, or (ii) r1 = −r2 = −r. It can be demonstrated that in the case (i) the surface f + is with necessity a M¨ obius transformation of f we will not consider this trivial case further. In the case (ii) one gets proper Darboux transformations. An important property of the Darboux transformations is the following: the quantity (1.76) c = 2 ss+ = |f + − f|2 ss+ is constant, i.e., does not depend on u ∈ R2. It is called a parameter of the Darboux transformation. The following data determine a Darboux trans- form f + of a given isothermic surface f uniquely: (D1) a point f + (0) (D2) a real number c, designated to be the constant (1.76). As usual, we regard iterating a Darboux transformation as adding a third (discrete) dimension to a two-dimensional isothermic net. The main classical result on Darboux transformations is the following theorem, which assures that one can add several discrete dimensions in a consistent way. Theorem 1.31. (Permutability of Darboux transformations) Let f be an isothermic surface, and let f (1) and f (2) be two of its Darboux trans- forms, with parameters c1 and c2, respectively. Then there exists a unique isothermic surface f (12) which is simultaneously a Darboux transform of f (1) with the parameter c2 and a Darboux transform of f (2) with the parameter c1. The surface f (12) is uniquely defined by the condition that the corre- sponding points of the four isothermic surfaces are concircular, and have a constant cross-ratio q ( f, f (1) , f (12) , f (2) ) = c1 c2 . Remark. To define the real cross-ratio of four concircular points a, b, c, d ∈ RN , one may identify the plane where they lie with the complex plane, and then set (1.77) q(a, b, c, d) = (a − b)(b − c)−1(c − d)(d − a)−1. In a more invariant fashion, one can use the Clifford algebra cross-ratio. For this aim, the points are interpreted as elements of the Clifford algebra C (RN ), and the above formula still holds, with all multiplication and in- version operations being performed in C (RN ), so that the ordering of the factors in this formula is essential.

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