20 1. Classical Differential Geometry These two circles intersect in two points. Such pairs of points comprise the two enveloping surfaces of the congruence, described in the M¨obius- geometric formalism as L4,1 ∩ ( span(ˆ s, ∂1ˆ ∂2ˆ) )⊥ . In other words, the en- velopes are represented by the elements ˆ ∈ L4,1 satisfying (1.65) ˆ ˆ = 0, ∂1ˆ ˆ = 0, ∂2ˆ ˆ = 0. Now, the principal parametrization is characterized by the following con- dition: when an infinitesimal displacement is made along one of the coor- dinate lines, say along the u2-line, the four points of contact of the two infinitely close spheres S(u), S(u + du2) with the envelopes lie on a circle, namely on C1(u). Indeed, differentiating the first two equations in (1.65) with respect to u2 and making use of the third and of equation (1.62), we come to ˆ ∂2ˆ = 0, ∂1ˆ ∂2ˆ = 0, which, compared with (1.63), demonstrates the claim. A convenient choice of representatives ˆ of hyperspheres S is the Eu- clidean one, in terms of the centers c and radii r: (1.66) ˆ : R2 → R 4,1 out ∩ {ξ0 = 1}, ˆ = c + e0 + ( |c|2 − r2 ) e∞. The condition for this to be a conjugate net in R4,1 leads to the following classical statement. Theorem 1.22. (Principally parametrized sphere congruences cen- ters and radii) A map (1.61) is a principally parametrized sphere congru- ence if and only if the centers c : R2 → R3 of the spheres S form a conjugate net in R3, and the radii r : R2 → R+ are such that the function |c|2 − r2 satisfies the same equation (1.1) as the centers c. 1.6. Surfaces with constant negative Gaussian curvature Up to now, we discussed special classes of coordinate systems in space, or special parametrizations of a general surface. Now, we turn to the discussion of several special classes of surfaces. The distinctive feature of these classes is the existence of transformations with certain permutability properties. One of the most prominent examples of integrability in differential geometry is given by the K-surfaces. Definition 1.23. (K-surface) An asymptotic line parametrized surface f : R2 → R3 is called a K-surface (or a pseudospherical surface) if its Gaussian curvature K is constant, i.e., does not depend on u ∈ R2. The following is their equivalent characterization as Chebyshev nets, i.e. nets with infinitesimal coordinate strips of constant width.

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