Delambre--Gauss Formulas in Complex Hyperbolic Space

Abstract

We establish a Delambre--Gauss matrix identity for oriented right-angled hexagons in the complex hyperbolic plane. The six complex length parameters of a hexagon determine two alternating \(3\times3\) matrices whose product is central in (\mathrm{SU}(2,1)\). The resulting nine entry-wise equations extend the Delambre-Gauss formula of Tan, Wong, and Zhang in 4-dimensional hyperbolic geometry to this complex hyperbolic setting.

arXiv comment22 pages, 2 figures