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.

28 pages, 4 figures; expanded Introduction, new Section, added references

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• Transformations of Quadrilateral Lattices
• Geometric Discretisation of the Toda System
• Factorisation of Macdonald polynomials
• The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
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Cited by in corpus (11)

• Desargues maps and the Hirota-Miwa equation
• Isothermic surfaces in sphere geometries as Moutard nets
• Discretization of asymptotic line parametrizations using hyperboloid patches
• Geometric discretization of the Koenigs nets
• Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices
• Discrete line complexes and integrable evolution of minors
• Generalized isothermic lattices
• On Weingarten transformations of hyperbolic nets
• Integrable discrete nets in Grassmannians
• Discrete Koenigs nets and discrete isothermic surfaces
• Bäcklund transformations as integrable discretization. The geometric approach