几何中保留的几何的特定PDE。III。一维射影变换与子群
Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
摘要：
通过求解特定的偏微分方程，我们将寻找几何上有意义的保持量扩展到1- $d$射影变换和子群。这不仅可以被看作是一个纯粹的几何问题，也可以被看作是寻找物理观测量的一个子案例，而且还可以被看作是将数学结构中的背景独立性的比较研究逐层扩展到包括投射结构的一部分。众所周知，全1- $d$投射不变量是交比。并将这一事实重新推导为1- $d$射影几何的保方程PDE系统的唯一解。我们还提供了1- $d$几何的保持量，其唯一的变换是1）特殊投影变换$Q$，给出倒数的差异。2)$Q$与膨胀$D$一起，现在给出倒数差的比率。此外，这种分析首先指出了对交比的一种新的解释：那些差异的比率同时是倒数的差异，其次促使1）和2）对应于真正的和独特的几何学。
英文摘要：
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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