Discrete asymptotic nets and W-congruences in Plucker line geometry

Abstract

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 Plucker 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-Backlund 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 Plucker. 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.