1.4. Orthogonal nets 17 The set of the touching spheres with this property (intersecting U in f only) has two connected components: M+ containing S(κ0) and M− containing S(−κ0) for big κ0 0. The boundary values κ1 = inf κ : S(κ) M+ , κ2 = sup κ : S(κ) M− are the principal curvatures of the surface in f. The directions in which S(κ1) and S(κ2) touch U are the principal directions. Curvature lines are integral curves of the principal directions fields. The symmetric functions K = κ1κ2, H = κ1 + κ2 2 are called the Gaussian curvature and the mean curvature, respectively. Clearly, all ingredients of this description are obius-invariant. Under a normal shift by the distance d the centers of the principal curvature spheres are preserved and their radii are shifted by d. This implies that the principal directions and thus the curvature lines are preserved under normal shifts, as well. A Lie-geometric nature of the curvature line parametrization yields that it has a Lie-invariant description. A surface in Lie geometry is considered as consisting of contact elements. A contact element can be identified with a pencil of spheres through a common point with a common (directed) normal in that point. Two infinitesimally close contact elements (sphere pencils) be- long to the same curvature line if and only if they have a sphere in common, which is the principal curvature sphere. Let us consider an infinitesimal neighborhood of a surface f with the Gauss map n. For sufficiently small t the formula ft = f + tn defines smooth surfaces parallel to f. The infinitesimal area of the parallel surface ft turns out to be a quadratic polynomial of t and is described by the classical Steiner formula (1.55) dA(ft) = (1 2Ht + Kt2)dA(f), Here dA is the infinitesimal area of the corresponding surface and H and K are the mean and the Gaussian curvatures of the surface f, respectively. 1.4.5. Ribaucour transformations of orthogonal nets. An important class of transformations between orthogonal nets is specified in the following definition. Definition 1.19. (Ribaucour transformation) A pair of m-dimensional orthogonal nets f, f + : Rm RN is related by a Ribaucour transformation if the corresponding coordinate curves of f and f + envelope one-parameter families of circles, i.e., if at every u Rm and for every 1 i m the
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