几何中保留的几何的特定PDE。II。仿射变换和子群
Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
摘要：
我们通过求解特定的偏微分方程来扩展寻找几何上显著的保持量到仿射变换和子群。这不仅可以看作是一个纯粹的几何问题，也可以看作是寻找物理可观量的一个子例，而且可以看作是数学结构中背景独立性逐层比较研究的一部分。虽然交叉和标量三重积（与差和比相结合）足以分别在2-和3- $d$中表达这些保留的量，但任意维的推广唤起了形式论。仿射保持量是差异的$d$ -体积形式的比率，$d$ -体积形式是由维度$d$支持的“顶部形式”，并且还涉及关系定义的子系统的$d$ -体积。
英文摘要：
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.
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