By identifying $\mathbb{R}^{2}$ with $\mathbb{C}$, we can think of the unit circle $\mathbb{S}^{1}$ as a subset of the complex plane. An angle function on a subset $U \subseteq \mathbb{S}^{1}$ is a continuous function $\theta: U \rightarrow \mathbb{R}$ such that $e^{i \theta(z)}=z$ for all $z \in U$. Show that there exists an angle function $\theta$ on an open subset $U \subseteq \mathbb{S}^{1}$ if and only if $U \neq \mathbb{S}^{1}$. For any such angle function, show that $(U, \theta)$ is a smooth coordinate chart for $\mathbb{S}^{1}$ with its standard smooth structure. (Used on $p p .37,152,176$.)

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Problem 1.8 (Angle function)

By identifying $ℝ^{2}$ with $ℂ$ , we can think of the unit circle $𝕊^{1}$ as a subset of the complex plane. An angle function on a subset $U \subseteq 𝕊^{1}$ is a continuous function $𝜃 : U \to ℝ$ such that $e^{i𝜃 (z)} = z$ for all $z \in U$ . Show that there exists an angle function $𝜃$ on an open subset $U \subseteq 𝕊^{1}$ if and only if $U \neq 𝕊^{1}$ . For any such angle function, show that $(U, 𝜃)$ is a smooth coordinate chart for $𝕊^{1}$ with its standard smooth structure. (Used on $pp . 37, 152, 176$ .)

Problem 1.8 (Angle function)

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