12 1. Classical Differential Geometry It can be demonstrated that the Lelieuvre normal fields of a Weingarten pair f, f + of A-surfaces satisfy (with the suitable choice of their signs) the following relation: (1.40) f + f = n+ × n. Differentiating the last equation and using the Lelieuvre formulas (1.38) for f and for f + , one easily sees that the normal fields of a Weingarten pair are related by (linear) differential equations: ∂1n+ + ∂1n = p1(n+ n), (1.41) ∂2n+ ∂2n = p2(n+ + n), (1.42) with some functions p1, p2 : R2 R. Thus: Theorem 1.14. (Weingarten transformation = Moutard transfor- mation for Lelieuvre normals) The Lelieuvre normal fields n, n+ of a Weingarten pair f, f + of A-surfaces are Moutard transforms of one another. A Weingarten transform f + of a given A-surface f is reconstructed from a Moutard transform n+ of the Lelieuvre normal field n. The data necessary for this are the data (MT1,2) for n: (W1) a point n+(0) R3 (W2) the values of the functions pi on the coordinate axes Bi for i = 1, 2, i.e., two smooth functions pi B i of one variable. The following statement is a direct consequence of Theorem 1.10. Theorem 1.15. (Permutability of Weingarten transformations) 1) Let f be an A-surface, and let f (1) and f (2) be two of its Weingarten transforms. Then there exists a one-parameter family of A-surfaces f (12) that are Weingarten transforms of both f (1) and f (2) . 2) Let f be an A-surface. Let f (1) , f (2) and f (3) be three of its Wein- garten transforms, and let three further A-surfaces f (12) , f (23) and f (13) be given such that f (ij) is a simultaneous Weingarten transform of f (i) and f (j) . Then generically there exists a unique A-surface f (123) that is a Wein- garten transform of f (12) , f (23) and f (13) . The net f (123) is uniquely defined by the condition that its every point lies in the tangent planes to f (12) , f (23) and f (13) at the corresponding points. 1.4. Orthogonal nets 1.4.1. Notion of orthogonal nets. An important subclass of conjugate nets is fixed in the following definition.
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