1.6. Surfaces with constant negative Gaussian curvature 21 Theorem 1.24. (K-surfaces = A-surfaces with Chebyshev prop- erty) An asymptotic line parametrized surface f : R2 → R3 is a K-surface if and only if the functions βi = |∂if| (i = 1, 2) depend on ui only: βi = βi(ui). One of the approaches to the analytical study of K-surfaces is based on the investigation of the angle φ(u1, u2) between asymptotic lines which is governed by the equation ∂1∂2φ = −Kβ1(u1)β2(u2) sin φ. After a re- parametrization of asymptotic lines one arrives at the famous sine-Gordon equation (1.67) ∂1∂2φ = sin φ. Another description is based on the Gauss maps. Theorem 1.25. (Gauss map of a K-surface is a Moutard net) The Lelieuvre normal field n : R2 → R3 of a K-surface with K = −1 takes values in the sphere S2 ⊂ R3, thus coinciding with the Gauss map. Conversely, any Moutard net in the unit sphere S2 is the Gauss map and the Lelieuvre normal field of a K-surface with K = −1. Moreover, |∂in| = βi (i = 1, 2), with the same functions βi = βi(ui) as in Theorem 1.24. Thus, the K-surfaces with K = −1 are in a one-to-one correspondence with the Moutard nets in S2. Functions n : R2 → S2 satisfying a Moutard equation (1.39) are sometimes called Lorentz-harmonic maps to S2 (one means hereby harmonicity with respect to the Lorentz metric on the plane R2 with coordinates (u1, u2)). It is important to observe that the coeﬃcient q12 of the Moutard equation (1.39) satisfied by a Lorentz-harmonic map n is completely determined by its first order derivatives: (1.68) q12 = ∂1∂2n, n = −∂1n, ∂2n . Therefore, the following data determine the Gauss map n of a K-surface f: (K) the values of the Gauss map on the coordinate axes B1, B2, i.e., two smooth curves n B i in S2 intersecting at a point n(0). The K-surface f is reconstructed from n uniquely, up to a translation, via formulas (1.38). Definition 1.26. (B¨ acklund transformation) A Weingarten transform f + of a K-surface f : R2 → R3 is called a B¨ acklund transform if the distance |f + − f| is constant, i.e., does not depend on u ∈ R2. As for a general Weingarten pair, the Lelieuvre normal fields (Gauss maps) n, n+ of a B¨ acklund pair of K-surfaces f, f + can be chosen so that eq. (1.40) holds, and hence n, n+ are related by the Moutard transformation (1.41), (1.42). From these equations it easily follows that for a B¨acklund pair the quantity n, n+ is constant thus, the intersection angle of the

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