On Discrete Differential Geometry in Twistor Space

TL;DR

This paper introduces a novel discrete integrable system in twistor space, extending classical differential geometry concepts to complex values and higher-dimensional spheres, with implications for discrete differential geometry.

Contribution

It generalizes the discrete cross-ratio system in $S^4$ to complex values using the Pl"ucker quadric, developing new geometric notions for discrete differential geometry in twistor space.

Findings

Defined discrete principal contact element nets for the Pl"ucker quadric

Proved elementary results generalizing classical Lie geometry

Connected results to previous work by Bobenko and Suris

Abstract

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Pl\"ucker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Pl\"ucker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discrete principal contact element nets for the Pl\"ucker quadric and prove several elementary results. Employing a second real real structure, we show that these results generalize previous results by Bobenko and Suris $(2007)$ on discrete differential geometry in the Lie quadric.