# Rigidity of Symmetric Frameworks on the Cylinder

## TL;DR

This paper studies the rigidity of symmetric structures on a cylinder, providing conditions for minimal rigidity under specific symmetries.

## Contribution

The paper introduces necessary and sufficient combinatorial conditions for symmetric frameworks on a cylinder to be isostatic.

## Key findings

- Necessary combinatorial conditions are given for symmetric frameworks on a cylinder to be isostatic.
- For cyclic symmetry groups, these conditions are sufficient under genericity assumptions.
- Symmetry is restricted to inversion, half-turn, or reflection symmetry in the analysis.

## Abstract

A bar-joint framework (G, p) is the combination of a finite simple graph \documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document}G=(V,E) and a placement \documentclass[12pt]{minimal}
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\begin{document}$$p:V\rightarrow {\mathbb {R}}^d$$\end{document}p:V→Rd. The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3$$\end{document}R3 restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.