18 1. Classical Differential Geometry straight lines spanned by the vectors ∂if, ∂if + at the respective points f, f + are interchanged by the reflection in the orthogonal bisecting hyperplane of the segment [f, f + ]. The net f + is called a Ribaucour transform of f. The nets f, f + serve as two envelopes of a Ribaucour sphere congruence Sm : Rm → {m-spheres in RN }. In other words, f(u), f + (u) ∈ Sm(u), and the tangent m-spaces to Sm(u) at f(u), resp. f + (u), are spanned by ∂if(u), resp. by ∂if + (u), i = 1, . . . , m. To describe a Ribaucour transformation analytically, we write: (1.56) ∂if + = ri ∂if − 2 ∂if, δf δf, δf δf , with some functions ri : Rm → R∗. It is convenient to define the metric coeﬃcients of the transformed net as h+ i = rihi = sign(ri)|∂if + |, with the corresponding unit vectors v+ i = (h+)−1∂if i + . Further, denote = |δf| and introduce the unit vector y = −1 δf, so that f + = f + y. Then we find: (1.57) v+ i = vi − 2 vi, y y, ∂iy = 1 2 θi(vi + + vi), with the functions θi : Rm → R defined as θi = (h+ i − hi)/ = (ri − 1)hi/. Equations (1.57) imply equations for the metric coeﬃcients: (1.58) h+ i = hi + θi, ∂i = −vi, y (h+ i + hi). Compatibility of the system (1.57) yields that θi have to satisfy certain differential equations: (1.59) β+ ij = βij − 2 vi, y θj, ∂iθj = 1 2 θi(βij + + βij). The following data determine a Ribaucour transform f + of a given orthog- onal net f uniquely: (R1) the point f + (0) (R2) the values of θi on the coordinate axes Bi for 1 ≤ i ≤ m, i.e., m smooth functions θi Bi of one variable. According to the general philosophy, iterating Ribaucour transforma- tions can be interpreted as adding an additional (discrete) dimension to an orthogonal net. The situation arising by adding two or three discrete dimensions is described in the following fundamental theorem. Theorem 1.20. (Permutability of Ribaucour transformations) 1) Let f be an m-dimensional orthogonal net, and let f (1) and f (2) be two of its Ribaucour transforms. Then there exists a one-parameter family of orthogonal nets f (12) that are Ribaucour transforms of both f (1) and f (2) . The corresponding points of the four orthogonal nets f, f (1) , f (2) and f (12) are concircular.

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