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 M¨ 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 M¨ 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 M¨ 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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.