1.3. Asymptotic nets 11 1.3. Asymptotic nets Definition 1.11. (A-surface) A map f : R2 → R3 is called an A-surface (an asymptotic line parametrized surface) if at every point the vectors ∂2f, 1 ∂2f 2 lie in the tangent plane to the surface f spanned by ∂1f, ∂2f. Thus, the second fundamental form of an A-surface in R3 is off-diagonal. Such a parametrization exists for a general surface with a negative Gaussian curvature. Definition 1.11, like the definition of conjugate nets, can be re- formulated so as to contain projectively invariant notions only. Therefore, A-surfaces actually belong to the geometry of the three-dimensional projec- tive space. In our presentation, however, we will use for convenience addi- tional structures on R3 (the Euclidean structure and the cross-product). A convenient description of A-surfaces is provided by the Lelieuvre representa- tion which states: there exists a unique (up to sign) normal field n : R2 → R3 to the surface f such that (1.38) ∂1f = ∂1n × n, ∂2f = n × ∂2n. Cross-differentiation of (1.38) reveals that ∂1∂2n×n = 0, that is, the Lelieu- vre normal field satisfies the Moutard equation (1.39) ∂1∂2n = q12n with some q12 : R2 → R. This reasoning can be reversed: integration of eqs. (1.38) with any solution n : R2 → R3 of the Moutard equation generates an A-surface f : R2 → R3. Theorem 1.12. (Lelieuvre normals of A-surfaces are Moutard nets) A-surfaces f : R2 → R3 are in a one-to-one correspondence (up to transla- tions of f) with Moutard nets n : R2 → R3, via the Lelieuvre representation (1.38). An A-surface f is reconstructed uniquely (up to a translation) from its Lelieuvre normal field n. In turn, a Moutard net n is uniquely determined by the initial data (M1,2), which we denote in this context by (A1,2): (A1) the values of the Lelieuvre normal field on the coordinate axes B1, B2, i.e., two smooth curves n Bi with a common intersection point n(0) (A2) a smooth function q12 : R2 → R that has the meaning of the coeﬃcient of the Moutard equation for n. Definition 1.13. (Weingarten transformation) A pair of A-surfaces f, f + : R2 → R3 is related by a Weingarten transformation if, for every u ∈ R2, the line ( f(u)f + (u) is tangent to both surfaces f and f + at the corresponding points. The surface f + is called a Weingarten transform of the surface f. The lines ( f(u)f + (u) ) are said to build a W-congruence.

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