16 1. Classical Differential Geometry Recall that we also have: hi = −∂iˆi, ˆ = −∂iˆi, ψ−1e0ψ , 1 ≤ i ≤ m. Thus, introducing vectors Si = ψˆiψ−1, we have the following expansion with respect to the vectors ek: (1.53) Si = ψˆiψ−1 = 1 2 ψ(∂iˆi)ψ−1 = − 1 2 k=i βkiek + hie∞. It is easy to see that (1.47) still holds, if the range of the indices is extended to all pairwise distinct i, j, k with 1 ≤ i, j ≤ m and 1 ≤ k ≤ N, and that the orthogonality constraint (1.48) can be now put as (1.54) ∂iβij + ∂jβji = − k=i,j βkiβkj . The system consisting of (1.47), (1.54) carries the name of the Lam´ e system. 1.4.4. Curvatures of surfaces and curvature line parametrized sur- faces. Two-dimensional (m = 2) orthogonal nets in R3 are nothing but sur- faces parametrized along curvature lines, or, otherwise said, parametrized so that both the first and the second fundamental forms are diagonal. Such a parametrization exists and is essentially unique for a general surface in R3 in the neighborhood of a nonumbilic point. In dimensions N 3 only special surfaces support such a parametrization. U κ2 0 κ1 0 n f Figure 1.1. Principal directions through touching spheres. Curvature lines are subject of Lie geometry, i.e., are invariant with re- spect to M¨ obius transformations and normal shifts. To see this, consider an infinitesimal neighborhood U of a point f of an oriented smooth surface in R3, and the pencil of spheres S(κ) with the curvatures κ, touching the surface at f see Figure 1.1. The curvature κ, as well as the signed radius r = 1/κ, is assumed positive if S(κ) lies on the same side of the tangent plane as the normal n, and negative otherwise the tangent plane itself is S(0). For big κ0 0 the spheres S(κ0) and S(−κ0) intersect U in f only.

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