Equivalence of Continuous, Local and Infinitesimal Rigidity in Normed Spaces

Sean Dewar

doi:10.1007/s00454-019-00135-5

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 1978/1979 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.

Highlights

A framework (G, p) is an embedding p of the vertices of a simple graph G into a given normed space
We wish to detect whether a framework in a general normed space is structurally rigid in the sense that either any continuous motion of the vertices that preserves the edge lengths corresponds to an isometric motion of the embedded vertices or that the embedding is locally unique up to an isometric map
We shall obtain further bounds on the dimension of the space of trivial infinitesimal motions for frameworks that lie on some hyperplane of the normed space. These results will allow us to prove that no small framework on two or more vertices is infinitesimally rigid in a non-Euclidean space

Summary

Introduction

A framework (G, p) is an embedding p of the vertices of a simple graph G into a given normed space. We will give an upper bound for the dimension of the space of trivial motions which will be achievable by most placements 5 we shall obtain further bounds on the dimension of the space of trivial infinitesimal motions for frameworks that lie on some hyperplane of the normed space. These results will allow us to prove that no small framework (a framework with less vertices than the dimension of the normed space plus one) on two or more vertices is infinitesimally rigid in a non-Euclidean space (see Theorem 5.8)

Preliminaries

Differentiation

Isometry Groups

Placements and Bar–Joint Frameworks

Structure of the Orbit of a Placement

Examples and Non-examples