Discrete asymptotic nets and W-congruences in Plücker line geometry

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plücker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W–congruences, which provide the Darboux–Bäcklund type transformation of asymptotic lattices. The permutability theorems for the discrete Moutard transformation and for the corresponding transformation of asymptotic lattices are established as well. Moreover it is proven that the discrete W–congruences are represented by quadrilateral lattices in the quadric of Plücker. These results generalize to a discrete level the classical line-geometric approach to asymptotic nets and W–congruences, and incorporate the theory of asymptotic lattices into more general theory of quadrilateral lattices and their reductions.