The rigidity of countable frameworks in normed spaces

Dewar, Sean and Kitson, Derek (2019) The rigidity of countable frameworks in normed spaces. PhD thesis, Lancaster University.

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Abstract

We present a rigorous study of framework rigidity in finite dimensional normed spaces using a wide array of tools to attack these problems, including differential and discrete geometry, matroid theory, convex analysis and graph theory. We shall first focus on giving a good grounding of the area of rigidity theory from a more general view point to allow us to deal with a variety of normed spaces. By observing orbits of placements from the perspective of Lie group actions on smooth manifolds, we obtain upper bounds for the dimension of the space of trivial motions for a framework. Utilising aspects of differential geometry, 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 establish the independence of all graphs with d + 1 vertices d-dimensional normed space, and also prove they will be flexible if the normed space is non-Euclidean. Next, we prove that a graph has an infinitesimally rigid placement in a nonEuclidean normed plane if and only if it contains a (2, 2)-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph K4 by considering smoothness and strict convexity properties of the unit ball. Finally, we carry our previous results to countably infinite frameworks where this is possible, and otherwise identify when such results cannot be brought forward. We first establish matroidal methods for identifying rigidity and flexibility, and apply these methods to a large class of normed spaces. We characterise a necessary and sufficient condition for countably infinite graphs to have sequentially infinitesimally rigid placements in a general normed plane, and further stengthen the result for a large class normed planes. Finally, we prove that infinitesimal rigidity for countably infinite generic frameworks implies a weaker (but possibly equivalent) form of continuous rigidity, and infinitesimal rigidity for countably infinite algebraically generic frameworks implies continuous rigidity.