14 1. Classical Differential Geometry nets can be consistently restricted to any quadric in a projective space. As we will see in Chapter 3, discrete differential geometry gives a clear insight into the origin of this nontrivial statement (and a simple proof). The quadric responsible for obius geometry is the light cone P(LN+1,1). Choosing various quadrics, we come to the classical geometries of Klein’s Erlangen program including the hyperbolic, spherical, Lie, Pl¨ ucker, Laguerre, etc. geometry. 1.4.2. Analytic description of orthogonal nets. For an analytic de- scription of an orthogonal net f : Rm RN , introduce metric coefficients hi = |∂if| and (pairwise orthogonal) unit vectors vi = h−1∂if. i Then the following equations hold: ∂if = hivi , (1.44) ∂ivj = βjivi , i = j, (1.45) ∂ihj = hiβij , i = j, (1.46) ∂iβkj = βkiβij , i = j = k = i, (1.47) which are analogous to (1.7)–(1.10). Indeed, equation (1.45) holds since f is a conjugate net and the vj are orthonormal, and it serves as a definition of rotation coefficients βji. Equation (1.46) is a direct consequence of (1.44), (1.45), while the Darboux system (1.47) expresses the compatibility of the linear system (1.45). So, one of distinctive features of orthogonal nets among general conjugate nets is that the system (1.4) admits a solution given by the locally defined metric coefficients hi. In the same spirit, the rotation coefficients βji reflect the local geometry of the net. The Darboux system (1.47) has to be supplemented by the orthogonality constraint (1.48) ∂iβij + ∂jβji = −∂ivi, ∂jvj , i = j. To derive (1.48), one considers the identity ∂i∂j vi, vj = 0 and makes use of (1.45). Equation (1.48) is an admissible constraint for the system (1.44)– (1.47). This is understood as follows: (1.48) involves two independent vari- ables i, j only, and therefore it makes sense to require that it be fulfilled on the coordinate plane Bij. One can easily check that if a solution to the system (1.44)–(1.47) satisfies (1.48) on all coordinate planes Bij for 1 i j m, then it is fulfilled everywhere on Rm. The meaning of the orthogonality condition (1.48) is that the coordinate surfaces f Bij are parametrized along curvature lines. The fact that this condition propagates is a sort of inversion of the classical Dupin theorem, which says that the coordinate surfaces of a triply orthogonal coordinate system intersect along their curvature lines. The coordinate surfaces f B ij can be characterized by the functions ηij = 1 2 (∂iβij ∂jβji) on Bij.
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