Geometric discretization of the Bianchi system

Abstract

We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion of discrete normal congruences. Finally, considering quadrilateral lattices "with equal tangential invariants" which allow for harmonic normal congruences we obtain, in complete analogy with the continuous case, the integrable discrete analogue of the Bianchi system together with its geometric meaning. To obtain this geometric meaning we also make use of the novel characterization of the circular lattice as a quadrilateral lattice whose coordinate lines intersect orthogonally in the mean.

arXiv comment26 pages, 7 postscript figures

References in corpus (9)

• Quantum Inverse Scattering Method and Correlation Functions
• Multidimensional Quadrilateral Lattices are Integrable
• Transformations of Quadrilateral Lattices
• Geometric Discretisation of the Toda System
• The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
• Quadratic reductions of quadrilateral lattices
• Geometric discretization of the Koenigs nets
• Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices
• The integrable discretization of the Bianchi-Ernst system

Cited by in corpus (7)

• Discrete differential geometry. Consistency as integrability
• The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction
• Geometric discretization of the Koenigs nets
• Integrable Systems and Discrete Geometry
• Generalized isothermic lattices
• Hirota equation and the quantum plane
• The normal dual congruences and the dual Bianchi lattice