单参数变换群

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定义 · 5.3 流与李导数

这个群称为单参数变换群(one-parameter group of transformations)。该群局部上看起来像加法群 $\mathbb{R}$，尽管它整体上未必同构于 $\mathbb{R}$。事实上，在例 5.9中，${\sigma }_{2\pi n+t}$ 与 ${\sigma }_t$ 是同一个映射，于是我们发现该单参数变换群同构于 SO(2)，即由形如 $2\times 2$ 的实矩阵所构成的乘法群，

$$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

英文原文

This group is called the one-parameter group of transformations. The group locally looks like the additive group $\mathbb{R}$ , although it may not be isomorphic to $\mathbb{R}$ globally. In fact, in example 5.9, ${\sigma }_{2\pi n+t}$ was the same map as ${\sigma }_t$ and we find that the one-parameter group is isomorphic to SO(2), the multiplicative group of $2\times 2$ real matrices of the form

$$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

出处：5.3 流与李导数 · node a-5eee5ab529f0df295e546189 · 源节点 SHA-256 8576b056a73b9ecf…