1.4. Orthogonal nets 15 Thus, the following data can be used to determine an orthogonal net f uniquely: (O1) the values of f on the coordinate axes Bi for 1 i m, i.e., m smooth curves f B i , intersecting pairwise orthogonally at f(0) (O2) m(m 1)/2 smooth functions ηij : Bij R for all 1 i j m. 1.4.3. Spinor frames of orthogonal nets. The obius-geometric de- scription of orthogonal nets has major conceptual and technical advantages. First, this description linearizes the invariance group of orthogonal nets, i.e., the obius group of the sphere SN (which can be considered as a compact- ification of RN by a point at infinity). Orientation preserving Euclidean motions of RN are represented as conjugations by elements of H∞, the isotropy subgroup of e∞ in Spin+(N + 1, 1). Further, using the Clifford algebra model of obius differential geometry enables us to give a frame description of orthogonal nets, which turns out to be a key technical device. As is easily seen, the metric coefficients hi = |∂if| satisfy also hi = |∂i ˆ|, where ˆ = f +e0 +|f|2e∞. Hence, the vectors ˆi = h−1∂ i i ˆ = vi +2 f, vi e∞ have the (Lorentz) length 1. Since ˆ ˆ = 0, one readily finds that ˆ ˆi = 0 and hi = −∂iˆi, ˆ . Theorem 1.18. (Spinor frame of an orthogonal net) For an orthogonal net f : Rm RN , i.e., for the corresponding conjugate net ˆ : Rm QN 0 , there exists a function ψ : Rm H∞ (called a frame of ˆ), such that ˆ = ψ−1e 0 ψ, (1.49) ˆi = ψ−1eiψ, 1 i m, (1.50) and satisfying the system of differential equations: (1.51) ∂iψ = −eiψˆi, ˆi = 1 2 ∂iˆi, 1 i m. Note that for an orthogonal coordinate system (m = N) the frame ψ is uniquely determined at any point by the requirements (1.49) and (1.50). It is readily seen that the unit tangent vectors ˆi satisfy eq. (1.45) with the same rotation coefficients βji = ∂iˆj, ˆi = −∂iˆi, ˆj . With the help of the frame ψ we extend the set of vectors {ˆi : 1 i m} to an orthonormal basis {ˆk : 1 k N} of T ˆ QN 0 : (1.52) ˆk = ψ−1ekψ, 1 k N. Respectively, we extend the set of rotation coefficients according to the for- mula βki = ∂iˆk, ˆi = −∂iˆi, ˆk = −∂iˆi, ψ−1ekψ , 1 i m, 1 k N.
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