Transformations of quadrilateral lattices

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x : Z −→ R , N ≤ M , whose elementary quadrilaterals are planar. Our investigation is based on the discrete analogue of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogues of the Laplace, Combescure, Lévy, radial and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂̄ formalism.