1.2. Koenigs and Moutard nets 7 Thus, p is a Combescure transform of f, and ψ is a function associated to φ, in Eisenhart’s terminology. Another computation leads to the relation (1.24) ∂iαj = cij(αi αj). The same argument as above shows that the data (F2) yield the values of φ+, and thus the values of αi, on the coordinate axes Bi. This uniquely specifies the functions αi everywhere on Rm as solutions of the compatible linear system (1.24) with the known coefficients cij. This, in turn, allows for a unique determination of the solutions p, ψ of equations (1.21), (1.22) with the initial data p(0) = f + (0) f(0) and ψ(0) = 1 (here the data (F1) enter into the construction). Thus, the classical formula (1.15) is recovered. 1.2. Koenigs and Moutard nets 1.2.1. Notion of Koenigs and Moutard nets. A geometrically impor- tant subclass of two-dimensional conjugate nets, very popular in the classical differential geometry, can be most directly defined as follows. Definition 1.5. (Koenigs net) A map f : R2 RN is called a Koenigs net if it satisfies a differential equation (1.25) ∂1∂2f = (∂2 log ν) ∂1f + (∂1 log ν) ∂2f with some scalar function ν : R2 R+. In other words, a Koenigs net is a two-dimensional conjugate net with the coefficients c21, c12 satisfying ∂1c21 = ∂2c12. Classically, this property has been interpreted as equality of the so-called Laplace invariants of the net (for this reason the Koenigs nets are also known as nets with equal invariants). Remarkably, this property is invariant under projective transformations, so that the notion of Koenigs nets actually belongs to projective geometry. The following data determine a Koenigs net f uniquely: (K1) the values of f on the coordinate axes B1, B2, i.e., two smooth curves f Bi with a common intersection point f(0) (K2) a smooth function ν : R2 R+. Leaving aside numerous geometric properties of Koenigs nets, discovered by the classics, we formulate here only the following characterization. Theorem 1.6. (Christoffel dual for a Koenigs net) A conjugate net f : R2 RN is a Koenigs net if and only if there exists a scalar function ν : R2 R+ such that the differential one-form df∗ defined by (1.26) ∂1f∗ = ∂1f ν2 , ∂2f∗ = ∂2f ν2
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