Quadrilateral and Circular Latticesare

This research is devoted to a systematic discretization of some classical notions of Diierential Geometry and to the study of their deep connections with the theory of integrable diierence equations in multidimensions. The basic example we consider here is the conjugate net 1], whose proper discretization is the quadrilateral lattice 2], 3]. are planar, where T i is the translation operator in the ith direction of the lattice. The planarity condition is the simplest linear constraint that can be imposed on the construction of the lattice; on each ij quadrilateral this condition is expressed by the following discrete analog of the Laplace equation whose compatibility gives the multidimensional quadrilateral lattice (MQL) equations: k A ij = (T j A jk)A ij + (T k A kj)A ik ? (T k A ij)A ik ; i 6 = j 6 = k 6 = i: (2) The construction of the lattice is expressed by the following boundary-value problem Proposition 1 3]Let us assign N arbitrary discrete intersecting curves in R M and