On Discrete Differential Geometry in Twistor Space

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 Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker 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 Plücker 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.

32 pages, 11 figures