Integrable discrete nets in Grassmannians

Abstract

We consider discrete nets in Grassmannians ${\mathbb{G}}_r^d$ which generalize Q-nets (maps ${\mathbb{Z}}^N\to {\mathbb{P}}^d$ with planar elementary quadrilaterals) and Darboux nets (${\mathbb{P}}^d$-valued maps defined on the edges of ${\mathbb{Z}}^N$ such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.

10 pp

References in corpus (3)

• On organizing principles of Discrete Differential Geometry. Geometry of spheres
• The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction
• Geometric algebra and quadrilateral lattices

Cited by in corpus (1)

• Billiard algebra, integrable line congruences, and double reflection nets