Polyhedra of Constant Gaussian Curvature

TL;DR

This paper demonstrates that in the discrete setting, any closed surface can be realized as a polyhedral surface with constant Gaussian curvature at all vertices, contrasting the rigidity seen in smooth surfaces.

Contribution

It proves that all closed surfaces can be embedded as polyhedral surfaces with uniform vertex curvature, revealing greater flexibility in the discrete setting.

Findings

Any closed surface can be realized with constant vertex curvature.

The result applies to orientable and non-orientable surfaces of any genus.

The proof is constructive and elementary.

Abstract

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface embedded in $\mathbb{R}^3$ with constant Gaussian curvature must be a sphere, reflecting the strong rigidity of differential geometry. In contrast, the discrete setting, where curvature is represented as an angular defect concentrated at vertices, admits far greater flexibility. For instance, a flat torus can be realised as a polyhedral surface in $\mathbb{R}^3$ with zero curvature at every vertex. We establish a general result: any closed surface, whether orientable or non-orientable and of arbitrary genus, can be realised in $\mathbb{R}^3$ as a (possibly self-intersecting) polyhedral surface in which every vertex has the same angular defect. This highlights a fundamental distinction between discrete and smooth settings, showing that curvature constraints in the discrete realm impose fewer restrictions. Our proof is constructive and, once recognised, entirely elementary. Yet this fundamental fact appears to have gone unnoticed in the existing literature.