1.1. Conjugate nets 5 One can iterate F-transformations and obtain a sequence f, f + , (f + )+, etc., of conjugate nets. We will see that this can be interpreted as generating a conjugate net of dimension M = m + 1, with m continuous directions and one discrete direction. The most remarkable property of F-transformations is the following permutability theorem. Theorem 1.3. (Permutability of F-transformations) 1) Let f be an m-dimensional conjugate net, and let f (1) and f (2) be two of its F-transforms. Then there exists a two-parameter family of conjugate nets f (12) that are F-transforms of both f (1) and f (2) . Corresponding points of the four conjugate nets f, f (1) , f (2) and f (12) are coplanar. 2) Let f be an m-dimensional conjugate net. Let f (1) , f (2) and f (3) be three of its F-transforms, and let three further conjugate nets f (12) , f (23) and f (13) be given such that f (ij) is a simultaneous F-transform of f (i) and f (j) . Then there exists generically a unique conjugate net f (123) that is an F-transform of f (12) , f (23) and f (13) . The net f (123) is uniquely defined by the condition that for every permutation (ijk) of (123) the corresponding points of f (i) , f (ij) , f (ik) and f (123) are coplanar. The situations described in this theorem can be interpreted as conjugate nets of dimension M = m + 2, resp. M = m + 3, with m continuous and two (resp. three) discrete directions. The theory of discrete conjugate nets allows one to put all directions on an equal footing and to unify the theories of smooth nets and of their transformations. Moreover, we will see that both these theories may be seen as a continuum limit (in some precise sense) of the fully discrete theory, if the mesh sizes of all or some of the directions become infinitely small (see Figure 0.1). This way of thinking is the guiding idea and the philosophy of the discrete differential geometry. The following special F-transformation is important in the surface the- ory. Definition 1.4. (Combescure transformation) We will say that two m- dimensional conjugate nets f, f + : Rm RN are related by a Combescure transformation if at every point u Rm and for each 1 i m the vectors ∂if, ∂if + are parallel. The net f + is called parallel to f, or a Combescure transform of the net f. 1.1.4. Classical formulation of F-transformation. Our formulation of F-transformations is rather different from the classical one, due to Jonas and Eisenhart, based on the formula (1.15) f + = f φ ψ p,
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