Adam Doliwa 2007 J. Phys. A: Math. Theor. 40 12539 doi:10.1088/1751-8113/40/42/S03
Adam Doliwa
Show affiliationsWe study multi-dimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Möbius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) cross-ratio of the circular lattice. We derive the analogous condition for our generalized isothermic lattices using Steiner's projective structure of conics, and we present basic geometric constructions which encode integrability of the lattice. In particular, we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two-dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.
35Q05 Euler-Poisson-Darboux equation and generalizations
15A63 Quadratic and bilinear forms, inner products [See mainly 11Exx]
Issue 42 (19 October 2007)
Received 15 January 2007, in final form 22 March 2007
Published 2 October 2007
Adam Doliwa 2007 J. Phys. A: Math. Theor. 40 12539
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