Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric

Abstract

Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$ upon quotienting out the similarity transformations. This theory is based on the geometry and topology of the corresponding configuration space: shape space. Kendall studied this to build a widely useful Shape Statistics thereupon. The corresponding Shape-and-Scale Theory -- quotienting out the Euclidean transformations -- is useful in Classical Dynamics and Molecular Physics, as well as for the relational `Leibnizian' side of the Absolute versus Relational Motion Debate. Kendall's shape spaces moreover recur withing this `Leibnizian' Shape-and-Scale Theory. There has recently been a large expansion in diversity of Kendall-type Shape(-and-Scale) Theories. The current article outlines this variety, and furthermore roots it in solving the poset of generalized Killing equations. This moreover also places a first great bound on how many more Shape(-and-Scale) Theories there can be. For it is nongeneric for geometrically-equipped manifolds -- replacements for Kendall's $\mathbb{R}^d$ carrier space (absolute space to physicists) - to possess any generalized Killing vectors. Article II places a second great bound, now at the topological level and in terms of which Shape(-and-Scale) Theories are technically tractable. Finally Article III explains how the diversity of Shape(-and-Scale) Theories - from varying which carrier space and quotiented-out geometrical automorphism group are in use - constitutes a theory of Comparative Background Independence: a topic of fundamental interest in Dynamics, Gravitation and Theoretical Physics more generally. Article I and II's great bounds moreover have significant consequences for Comparative Background Independence.