1.4. Orthogonal nets 13 Definition 1.16. (Orthogonal net) A conjugate net f : Rm RN is called an m-dimensional orthogonal net in RN if at every u Rm and for all pairs 1 i = j m we have ∂if ∂jf . Such a net is called an orthogonal coordinate system if m = N. The class of orthogonal nets (as well as their Ribaucour transformations see Definition 1.19 below) are invariant under obius transformations and therefore belong to obius differential geometry. To demonstrate this, it is enough to show the invariance with respect to the inversion f ˜ = f/|f|2. A direct computation shows that the inversion maps a conjugate net with the coefficients cij and with the orthogonality property to a conjugate net with the coefficients ˜ij = cij 2 ∂if, f /|f|2, which is orthogonal again. Since orthogonal nets belong to obius differential geometry, it is useful to describe them with the help of the corresponding apparatus (a sketch of which is given in Section 9.3). In this formalism, the points of RN (or, better, of the conformal N-sphere SN , which is a compactification of RN ) are represented by elements of the projectivized light cone P(LN+1,1) in the projectivized Minkowski space P(RN+1,1). The light cone LN+1,1 = ξ RN+1,1 : ξ, ξ = 0 is of central importance in obius geometry (the absolute quadric). Let {e1, . . . , eN+2} denote the standard basis of the Minkowski space RN+1,1. We denote also e0 = 1 2 (eN+2 eN+1) and e∞ = 1 2 (eN+2 + eN+1). The Euclidean space RN is identified, via (1.43) π0 : RN f ˆ = f + e0 + |f|2e∞ QN 0 , with the section QN 0 of the cone LN+1,1 by the affine hyperplane {ξ0 = 1}, where ξ0 is the e0-component of ξ RN+1,1 in the basis {e1, . . . , eN , e0, e∞}. An elegant characterization of orthogonal nets is due to Darboux: Theorem 1.17. (M¨ obius-geometric characterization of orthogonal nets) A conjugate net f : Rm RN is orthogonal if and only if the scalar function |f|2 satisfies the same equation (1.1) as f does, or, equivalently, if the lift ˆ = π0 f : Rm QN 0 is a conjugate net in RN+1,1. In other words, the image of an orthogonal net in the projectivized light cone P(LN+1,1) is a conjugate net in P(RN+1,1). In particular, any lift ˜ = λ ˆ of f in LN+1,1, not necessarily normalized as in (1.43), satisfies linear differential equations (1.3). This criterion makes the invariance of orthogo- nal nets under obius transformations self-evident. It will be important to preserve this symmetry group under discretization. This deep result by Darboux is an instance of a very general phenomenon which will be used many times within this book. It turns out that conjugate
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