6 1. Classical Differential Geometry whose data are: an additional solution φ : Rm → R of (1.1), a Combescure transform p : Rm → RN of f, and the function ψ : Rm → R, associated to φ in the same way as p is related to f. We now demonstrate how to identify these ingredients within our approach and how they are specified by the initial data (F1,2). It follows from (1.12)–(1.14) that (1.16) ∂i bj aj = cij bj aj + cji bi ai − bjbi ajai . The symmetry of the right-hand sides of (1.16), (1.13) yields the existence of the functions φ, φ+ : Rm → R such that (1.17) ∂iφ φ = bi ai , ∂iφ+ φ+ = bi , 1 ≤ i ≤ m. These equations define φ, φ+ uniquely up to respective constant factors, which can be fixed by requiring φ(0) = φ+(0) = 1. An easy computation based on (1.16), (1.13) shows that the functions φ, φ+ satisfy the following equations: ∂i∂jφ = cij∂jφ + cji∂iφ, (1.18) ∂i∂jφ+ = c+∂jφ+ ij + c+∂iφ+, ji (1.19) for all 1 ≤ i = j ≤ m. Thus, an F-transformation yields some additional scalar solutions φ and φ+ of the equations describing the nets f and f + , respectively. Of these two, the solution φ is directly specified by the original net f and the initial data (F2). Indeed, the data (F2) yield the values of φ along the coordinate axes, through integrating the first equations in (1.17) these values determine the solution of (1.18) with the known coefficients cij uniquely. Further, introduce the quantities (1.20) p = f + − f φ+ , ψ = − φ φ+ . Then a direct computation based on (1.11), (1.12)–(1.14), and (1.17) shows that the following equations hold: ∂ip = αi∂if, (1.21) ∂iψ = αi∂iφ, (1.22) where (1.23) αi = ai − 1 φ+ .
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