Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups

Abstract

We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical observables, and furthermore as part of the comparative study of Background Independence level-by-level in mathematical structure. While cross and scalar-triple products (combined with differences and ratios) suffice to formulate these preserved quantities in 2- and 3-$d$ respectively, the arbitrary-dimensional generalization evokes the theory of forms. The affine preserved quantities are ratios of $d$-volume forms of differences, $d$-volume forms being the `top forms' supported by dimension $d$, and referring moreover to $d$-volumes of relationally-defined subsystems.

15 pages, including 3 figures. Updated references

References in corpus (7)

• Some remarks on the notions of general covariance and background independence
• Nonparametric statistics on manifolds with applications to shape spaces
• Problem of Time and Background Independence: the Individual Facets
• A Local Resolution of the Problem of Time
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups

Cited by in corpus (7)

• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Geometry from Brackets Consistency
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables