8 1. Classical Differential Geometry is closed. In this case the map f∗ : R2 RN , defined (up to a translation) by the integration of this one-form, is also a Koenigs net, called Christoffel dual to f. This follows immediately by cross-differentiating (1.26). A different way to formulate the latter equations is: ∂1f∗ ∂1f, ∂2f∗ ∂2f, (∂1 + ∂2)f∗ (∂1 ∂2)f, (∂1 ∂2)f∗ (∂1 + ∂2)f. (1.27) If one considers the ambient space RN of a Koenigs net as an affine part of RPN , then there is an important choice of representatives for f (f, 1) in the space RN+1 of homogeneous coordinates, namely (1.28) y = ν−1(f, 1). Indeed, a straightforward computation shows that the representatives (1.28) satisfy the following simple differential equation: (1.29) ∂1∂2y = q12y with the scalar function q12 = ν∂1∂2(ν−1). Differential equation (1.29) is known as the Moutard equation and y is called a Moutard representative of the Koenigs net f. Definition 1.7. (Moutard net) A map y : R2 RN+1 is called a Moutard net if it satisfies the Moutard differential equation (1.29) with some q12 : R2 R. Thus, we see that Moutard nets appear as special lifts of Koenigs nets to the space of homogeneous coordinates. Conversely, if y is a Moutard net in RN+1, then it is not difficult to figure out the condition for a scalar function ν : R2 R, under which ˜ = νy satisfies an equation of the type (1.1): ν−1 has to be a solution of the same Moutard equation (1.29), and then ∂1∂2 ˜ = (∂2 log ν)∂1 ˜+ (∂1 log ν)∂2 ˜ For instance, one can choose ν−1 to be any component of the vector y in this case the N components of ˜ = νy which are different from 1 build a Koenigs net in RN . Of course, Moutard nets can be considered also in their own right, i.e., one does not have to regard the ambient space RN+1 of a Moutard net as the space of homogeneous coordinates for RPN . Nevertheless, such an interpretation is useful in most cases. The following data determine a Moutard net y uniquely: (M1) the values of y on the coordinate axes B1, B2, i.e., two smooth curves y Bi with a common intersection point y(0)
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