The Normal Dual Congruences and the Dual Bianchi Lattice

The main goal of the paper is to find the discrete analogue of the Bianchi system in spaces of arbitrary dimension together with its geometric interpretation. We show that the proper geometric framework of such generalization is the language of dual quadrilateral lattices and of dual congruences. After introducing the notion of the dual Koenigs lattice in a projective space of arbitrary dimension we define the discrete dual congruences and we present, as an important example, the normal dual discrete congruences. Finally, we introduce the dual Bianchi lattice as a dual Koenigs lattice allowing for a conjugate normal dual congruence, and we find its characterization in terms of a system of integrable difference equations.