几何中保留的几何的特定PDE。I.相似性和子群
Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
摘要：
我们为几何学中的守恒量$Q$提供了特定的偏微分方程，并为物理学中的可观测量$O$提供了这一偏微分方程和特定偏微分方程之间的桥梁。我们进一步证明了这座桥两侧的其他四个定理的版本：下面列举的句子。对于一般几何-在它不拥有广义Killing向量的意义上，即连续几何自同构--$P$在所述几何上形成自由函数的光滑空间。如果一个几何拥有相应类型的Killing向量，则$P$必须与“自同构生成元的点求和”$S$进行李括号交换。与此对应的可观的是，在存在第一类约束$F$的情况下，$O$必须与这些约束泊松括号交换。则1）定义$Q$，$O$需要$S$，$F$的闭子代数。2)$Q$和$O$本身形成闭代数。3)$Q$，$O$的子代数分别与$S$，$F$的子代数形成对偶的有界格.此外，$S$，$Q$和$F$，$O$交换都可以被重新表示为一阶线性偏微分方程，被自由特征地处理。第二个最通用的情况只有一个$S$或$F$，因此只有一个PDE，标准地简化为ODE系统。然而，多个$S$或$F$的更高度非通用的情况返回超定PDE系统。4)我们证明，尽管如此，这些总是可积的。这一点很重要，因为它与我们更熟悉的广义Killing方程本身的行为相反。我们终于解决了保存数量的相似几何及其子群;同伴文件扩展到仿射，射影和共形几何。
英文摘要：
We provide specific PDEs for preserved quantities $Q$ in Geometry, as well as a bridge between this and specific PDEs for observables $O$ in Physics. We furthermore prove versions of four other theorems either side of this bridge: the below enumerated sentences. For the generic geometry - in the sense of it possessing no generalized Killing vectors, i.e.\ continuous geometrical automorphisms - the $P$ form a smooth space of free functions over said geometry. If a geometry possesses the corresponding type of Killing vectors, the $P$ must Lie-brackets commute with `sums-over-points of the automorphism generators', $S$. The observables counterpart of this is that in the presence of first-class constraints $F$, the $O$ must Poisson-brackets commute with these. Then 1) defining $Q$, $O$ requires closed subalgebras of $S$, $F$. 2) The $Q$, and the $O$, themselves form closed algebras. 3) The subalgebras of $Q$, $O$ form bounded lattices dual to those of $S$, $F$ respectively. Both $S$, $Q$ and $F$, $O$ commutations can moreover be reformulated as first-order linear PDEs, treated free-characteristically. The secondmost generic case has just one $S$ or $F$, and so just one PDE, which standardly reduces to an ODE system. The more highly nongeneric case of multiple $S$ or $F$, however, returns an over-determined PDE system. 4) We prove that nonetheless these are always integrable. This is significant by being mostly-opposite to how the more familiar generalized Killing equations themselves behave. We finally solve for the preserved quantities of similarity geometry and its subgroups; companion papers extend this program to affine, projective and conformal geometries.
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