note-math

$𝑝$ 附近的向量场 $𝑌$, 在 $𝑋$ 方向, 尝试在坐标里求导数 $\frac{\partial 𝑌}{\partial 𝑋}=\underset{𝑡\to 0}{\text{ lim }}\frac{1}{𝑡}\left(𝑌\left(𝑝+𝑡𝑋\right)-𝑌\left(𝑝\right)\right)$

然而, #link(<difference-polynomial>)[difference] 操作不线性兼容于一般 diffeomorphism 的换坐标

但是在 metric-manifold, 有特殊的坐标 — #link(<geodesic-coordinate>)[测地线坐标]. $𝑝$ 的不同测地线坐标的变换方式是 $\text{SO}$, 是线性的

在 $𝑝$ 点测地线坐标, 在 $𝑝$ 点的导数, ${\nabla }_{𝑋}𝑌≔\frac{\partial 𝑌}{\partial 𝑋}$

也可以对 #link(<tensor>)[] 场求导数 ${\nabla }_{𝑋}𝑇$. 根据张量结构带有的纯量乘法, 计算可以使用 #link(<Leibniz-law>)[product-rule] Example ${\nabla }_{𝑋}\left(𝑌\otimes 𝑍\right)=\left({\nabla }_{𝑋}𝑌\right)\otimes 𝑍+𝑌\otimes \left({\nabla }_{𝑋}𝑍\right)$

用一般坐标 $𝑥$ 计算出测地线坐标 $𝑦$, 然后在坐标 $𝑦$, 测地线导数是

$$\begin{array}{ccc}{\nabla }_{𝑋}𝑌\left(𝑥\right)& =& \frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)\cdot \frac{\partial 𝑌}{\partial 𝑋}\left(𝑦\right)\\ & =& \frac{\partial }{\partial 𝑋}\left(\frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌\right)-\frac{\partial }{\partial 𝑋}\left(\frac{\partial 𝑥}{\partial 𝑦}\right)\cdot 𝑌& & \text{ by product-rule}\end{array}$$

使用 #link(<connection-transformation>)[联络的变换]

$$\begin{array}{ccc}0& =& \mathrm{\Gamma }\left(𝑦\right)\\ & =& \frac{\partial 𝑦}{\partial 𝑥}\cdot \mathrm{\Gamma }\left(𝑥\right)\cdot \frac{\partial 𝑥}{\partial 𝑦}+\frac{\partial 𝑦}{\partial 𝑥}\cdot \frac{\partial }{\partial 𝑦}\left(\frac{\partial 𝑥}{\partial 𝑦}\right)\end{array}$$

==> $\frac{\partial }{\partial 𝑦}\left(\frac{\partial 𝑥}{\partial 𝑦}\right)=-\mathrm{\Gamma }\left(𝑥\right)\cdot \frac{𝑑𝑥}{𝑑𝑦}$

使用 $\frac{\partial }{\partial 𝑋}\left(\frac{\partial 𝑥}{\partial 𝑦}\right)=\frac{\partial }{\partial 𝑦}\left(𝑋\right)\left(\frac{\partial 𝑥}{\partial 𝑦}\right)$. 代入 ${\nabla }_{𝑋}𝑌\left(𝑦\right)$ 的计算

$${\nabla }_{𝑋}𝑌\left(𝑦\right)=\frac{\partial }{\partial 𝑋}\left(\frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌\right)+{𝑋}^{⊺}\cdot \mathrm{\Gamma }\left(𝑥\right)\cdot \frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌$$

$𝑝$ 切空间将 ${\nabla }_{𝑋}𝑌\left(𝑦\right)$ 线性转换 $\frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)$ 到 ${\nabla }_{𝑋}𝑌\left(𝑥\right)$, 但保持 in coordinate $𝑥$, but keep $𝑋,𝑌$ in coordinate $𝑦$

$$\begin{array}{cc}& {\nabla }_{𝑋}𝑌\left(𝑥\right)\\ & =& \frac{\partial }{\partial \left(\frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)\cdot 𝑋\left(𝑦\right)\right)}\left(\frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)\cdot 𝑌\left(𝑦\right)\right)+{\left(\frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)\cdot 𝑋\left(𝑦\right)\right)}^{⊺}\cdot \mathrm{\Gamma }\left(𝑥\right)\cdot \frac{\partial 𝑥}{\partial 𝑦}\left(𝑦\right)\cdot 𝑌\left(𝑦\right)\\ & =& \frac{\partial }{\partial \left(𝑋\left(𝑥\right)\right)}𝑌\left(𝑥\right)+{𝑋\left(𝑥\right)}^{⊺}\cdot \mathrm{\Gamma }\left(𝑥\right)\cdot 𝑌\left(𝑥\right)\end{array}$$

或者写为, 在一般坐标, 测地线导数

$$\nabla =\partial +\mathrm{\Gamma }$$

对于 coordinate-frame

$${\nabla }_{\frac{\partial }{\partial {𝑥}^{𝑖}}}\frac{\partial }{\partial {𝑥}^{{𝑖}^{\prime }}}={\mathrm{\Gamma }}_{𝑖{𝑖}^{\prime }}^{𝑗}\frac{\partial }{\partial {𝑥}^{𝑗}}$$

有无更直观的解释, 而不是直接使用联络的变换?

如果只看线性兼容, 那么有很多 #link(<principal-bundle-connection>)[线性 connection], 重合于 geodesic-derivative 的是 metric-connection

Question 类似于 vector 场的情况. 使用变换 $𝛼\left(𝑥\right)=𝛼\left(𝑦\right)\cdot \frac{\partial 𝑥}{\partial 𝑦}$ 和 product-rule $\frac{\partial }{\partial 𝑋}\left(𝛼\cdot \frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌\right)=\frac{\partial 𝛼}{\partial 𝑋}\cdot \left(\frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌\right)+𝛼\cdot \frac{\partial }{\partial 𝑋}\left(\frac{\partial 𝑥}{\partial 𝑦}\cdot 𝑌\right)$

对于 co-vector coordinate-frame

$${\nabla }_{\frac{\partial }{\partial {𝑥}^{𝑖}}}𝑑{𝑥}^{{𝑖}^{\prime }}=-{\mathrm{\Gamma }}_{𝑖𝑗}^{{𝑖}^{\prime }}𝑑{𝑥}^{𝑗}$$

平行运输 as "沿曲线零变化率" ${\nabla }_{\dot{𝑥}}𝑋=0$ or ${\left(\partial +\mathrm{\Gamma }\right)}_{\dot{𝑥}}𝑋=0$ where $𝑋=𝑋\left(𝑥\left(𝑡\right)\right)$

${\nabla }_{\dot{𝑥}}𝑋=0$ 是 ODE

根据计算 (?) 可以从平行运输 + 微商恢复协变导数

metric-connection 的平行运输保持 metric

可以用来构造规范正交标架

可以证明流形 metric 一一对应的到流形上的 $\text{SO}$ principal-bundle 结构

但是有更具体且可操作的计算结果吗? 关于在测地线坐标用平行运输计算规范正交标架

规范正交标架可能会用于弯曲流形的 spinor 的一些简化计算 e.g. Pauli-matrix ${𝜎}^{0,1,2,3}$