10 1. Classical Differential Geometry Definition 1.9. (Koenigs transformation) Two Koenigs nets f, f + : R2 RN are said to be related by a Koenigs transformation if the focal points F (1) , F (2) of the line congruence (ff + ) separate the points f, f + harmonically. It can be shown that any Koenigs transformation is analytically repre- sented as the Moutard transformation (1.30), (1.31) by a suitable choice of Moutard lifts y, y+. Theorem 1.10. (Permutability of Moutard transformations) 1) Let y be a Moutard net, and let y(1) and y(2) be two of its Moutard transforms. Then there exists a one-parameter family of Moutard nets y(12) that are Moutard transforms of both y(1) and y(2). 2) Let y be a Moutard net. Let y(1), y(2) and y(3) be three of its Moutard transforms, and let three further Moutard nets y(12), y(23) and y(13) be given such that y(ij) is a simultaneous Moutard transform of y(i) and y(j). Then generically there exists a unique Moutard net y(123) that is a Moutard trans- form of y(12), y(23) and y(13). 1.2.3. Classical formulation of the Moutard transformation. Due to the first equation in (1.32), for any Moutard transformation there exists a function θ : R2 R, unique up to a constant factor, such that (1.35) p1 = ∂1θ θ , p2 = ∂2θ θ . The last equation in (1.32) implies that θ satisfies (1.29). This scalar solution of (1.29) can be specified by its values on the coordinate axes Bi (i = 1, 2), which are readily obtained from the data (MT2) by integrating the corresponding equations (1.35). This establishes a bridge to the classical formulation of the Moutard transformation, according to which a Moutard transform y+ of the solution y of the Moutard equation (1.29) is specified by an additional scalar solution θ of this equation, via (1.30), (1.31) with (1.35). Note that these equations can be equivalently rewritten as (1.36) ∂1(θy+) = −θ2∂1 y θ , ∂2(θy+) = θ2∂2 y θ . From these equations one can conclude that y+ solves the Moutard equation (1.29) with the transformed potential (1.37) q+ 12 = q12 2∂1∂2 log θ = ∂1∂2θ+ θ+ , θ+ = 1 θ . In our formulation, the origin of the function θ becomes clear: it comes from p1, p2 by integrating the system (1.35). Equation (1.37) is then nothing but an equivalent form of (1.33).
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