Analytic curves are classified w.r.t. their symmetry under a given regular and separately analytic Lie group action $G\times M\to M$ on an analytic manifold. We show that a non-constant analytic curve $\gamma :D\to M$ is either free or exponential – i.e., up to analytic reparametrization of the form $t\mapsto \exp (t\cdot \overrightarrow{g})\cdot x$. The vector $\overrightarrow{g}\in \mathfrak{g}$ is additionally proven to be unique up to (non-zero scalation and) addition of elements in the Lie algebra of the stabilizer $G_{\gamma }\equiv \{g\in G|g\cdot \gamma =\gamma \}$ of the curve γ. We furthermore prove that in the free case, γ splits into countably many immersive subcurves – each of them discretely generated by G. This means that each such subcurve $\delta :D\supseteq ({\iota }^{\prime },\iota )\to M$ is build up countably many G-translates of a symmetry free building block $\delta {|}_{\mathrm{\Delta }}$, whereby three different cases can occur:

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In the shift case, the building blocks are continuously distributed in δ, with Δ always compact. Then, δ is created by iterated shifts of $\delta {|}_{\mathrm{\Delta }}$ by some $g\in G$ and its inverse; whereby the class $[e]\ne [g]\in G/G_{\gamma }$ is uniquely determined, as well as the same for each possible decomposition.

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In the flip case, there exist countably many building blocks – each of them defined on a compact interval, and contained in the one and only decomposition that exists in this case. Here, δ is created by iterated flips at the boundary points of these building blocks, whereby the occurring transformations are generated by two non-trivial classes in $G/G_{\gamma }$.

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In the mirror case, there exists exactly one symmetry (flipping) point $\delta (\tau )$, as well as one translation class $[e]\ne [g]\in G/G_{\gamma }$. The one and only decomposition of δ is thus given by $\delta {|}_{(i^{\prime },\tau ]}$, $\delta {|}_{[\tau ,i)}$, whereby $\delta {|}_{(i^{\prime },\tau ]}$ is flipped into $\delta {|}_{[\tau ,i)}$ or vice versa (or both).

在给定的规则分析李群作用下，分析曲线根据其对称性进行分类 $G\times \mathrm{中号}\to \mathrm{中号}$在分析流形上。我们证明了一个非恒定的分析曲线$\gamma ：d\to \mathrm{中号}$ 是自由的还是指数的–即，直到形式的解析重新参数化 $Ť\mapsto \mathrm{经验值}（Ť\cdot \overrightarrow{G}）\cdot X$。向量$\overrightarrow{G}\in \mathfrak{G}$ 还被证明是唯一的，直到（非零缩放和）稳定器的李代数中的元素加法 $G_{\gamma }\equiv \{G\in G|G\cdot \gamma =\gamma \}$曲线的γ。我们进一步证明，在自由情况下，γ分成许多沉浸式子曲线，每个子曲线由G离散生成。这意味着每个这样的子曲线$\delta ：d\supseteq （{\iota }^{\prime }，\iota ）\to \mathrm{中号}$建立了许多无对称构件的G-平移$\delta {|}_{\mathrm{\Delta }}$，因此可能发生三种不同情况：

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在换档的情况下，构建块以δ连续分布，而Δ始终紧凑。然后，δ是通过迭代移位创建$\delta {|}_{\mathrm{\Delta }}$ 由一些 $G\in G$及其逆 从而上课$[Ë]\ne [G]\in G/G_{\gamma }$ 唯一确定，每个可能的分解也相同。

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在翻转情况下，存在无数个构造块–每个构造块都在一个紧凑的间隔内定义，并且包含在这种情况下唯一存在的分解中。此处，δ是通过在这些构造块的边界点处进行迭代翻转而创建的，其中，发生的变换是由两个非平凡的类生成的$G/G_{\gamma }$。

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在镜像情况下，仅存在一个对称（翻转）点 $\delta （\tau ）$以及一个翻译班 $[Ë]\ne [G]\in G/G_{\gamma }$。因此，δ的唯一分解为$\delta {|}_{（{\mathrm{一世}}^{\prime }，\tau ]}$， $\delta {|}_{[\tau ，\mathrm{一世}）}$， $\delta {|}_{（{\mathrm{一世}}^{\prime }，\tau ]}$ 被翻转成 $\delta {|}_{[\tau ，\mathrm{一世}）}$ 反之亦然（或两者都有）。