Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces

TL;DR

This paper explores the conditions under which graph frameworks are globally rigid in complex and pseudo-Euclidean spaces, revealing equivalences and properties that extend classical rigidity results to these non-Euclidean settings.

Contribution

It establishes that a graph's generic global rigidity in Euclidean space is equivalent to its rigidity in complex and pseudo-Euclidean spaces, and provides conditions for generic rigidity in these spaces.

Findings

Global rigidity in Euclidean space iff in complex or pseudo-Euclidean space

Global rigidity is a generic property in complex space

Provides sufficient conditions for pseudo-Euclidean space rigidity

Abstract

In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space ($R^d$ with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.