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arXiv · 1809.02065

Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups

Edward Anderson
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Abstract

We extend finding geometrically-significant preserved quantities by solving specific PDEs to 1-$d$ projective 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 extending the comparative study of Background Independence level-by-level in mathematical structure to include projective structure. Full 1-$d$ projective invariants are well-known to be cross-ratios. We moreover rederive this fact as the unique solution of 1-$d$ projective geometry's preserved equation PDE system. We also provide the preserved quantities for the 1-$d$ geometries whose only transformations are 1) special-projective transformations $Q$, giving differences of reciprocals. 2) $Q$ alongside dilations $D$, now giving ratios of difference of reciprocals. This analysis moreover firstly points to a new interpretation of cross-ratio: those ratios of differences that are concurrently differences of reciprocals, and secondly motivates 1) and 2) as corresponding to bona fide and distinctive Geometries.

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Edward Anderson. 2018-09-06. Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups. https://arxiv.org/abs/1809.02065

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