Equivalence of continuous, local and infinitesimal rigidity in normed spaces

TL;DR

This paper investigates the concept of framework rigidity across various normed spaces, extending classical Euclidean results and providing new bounds and flexibility results in non-Euclidean contexts.

Contribution

It generalizes the equivalence of local, continuous, and infinitesimal rigidity to all finite-dimensional normed spaces and introduces bounds on trivial motions and flexibility in non-Euclidean spaces.

Findings

Proves equivalence of rigidity notions in general normed spaces.

Provides upper bounds for trivial motion space dimensions.

Shows flexibility of small frameworks in non-Euclidean spaces.

Abstract

We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth's 1978/9 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we obtain upper bounds for the dimension of the space of trivial motions for a framework and establish the flexibility of small frameworks in general non-Euclidean normed spaces.