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2020-07-12 01:48:31 +0100 asked a question Defining $S^1$ with four charts


I am trying to implement the construction of the manifold $S^1$ using the four charts, as described at . It's not clear to me how to define the charts in terms of the projection operators within SageMath. I'm also not sure how to isolate open sets that I would like to define the charts over. The circle is parameterized using polar coordinates, and the four charts have the form:

$ \begin{align} \varphi_1(r, \theta) &= \theta  && \theta \in (-\frac{\pi}{3}, \frac{\pi}{3}) \ \tag{1} \end{align} $

$ \begin{align} \varphi_2(r, \theta) &= \theta  && \theta \in (\frac{\pi}{6}, \frac{5\pi}{6}) \ \tag{2} \end{align} $

$ \begin{align} \varphi_3(r, \theta) &= \theta  && \theta \in (\frac{2\pi}{3}, \frac{4\pi}{3}) \ \tag{3} \end{align} $

$ \begin{align} \varphi_4(r, \theta) &= \theta  && \theta \in (\frac{7\pi}{6}, \frac{11\pi}{6}) \ \tag{4} \end{align} $

Once we have these charts, how would I define the four transition functions to complete the description?

Thank you for your help!