[Math] Only one chart to parameterize a unit-cylinder


I want to parameterize a unit-cylinder $x^2+y^2=1$ with only one chart in a complete atlas (the sets must be open). The cylinder is in $\mathbb{R}^3$. One way to do the parametrization with two charts is:
$$ \textbf{x}(u,v)=(\cos u, \sin u, v)$$
with $v\in \mathbb{R}$ and $0<u< 2\pi$ (this is the open set $U_1=(0,2\pi)\times \mathbb{R}$) and
$$ \textbf{y}(\overline{u}, \overline{v}) = (\cos \overline{u}, \sin \overline{u}, \overline{v})$$
with $\overline{v}\equiv v\in \mathbb{R}$ and $-\pi < \overline{u} < \pi$ (this is the open set $U_2 = (-\pi,\pi)\times\mathbb{R}$). So the atlas is $\{ \textbf{x}, \textbf{y}\}$ into $U_1 \times U_2.$ So, it has two charts $\textbf{x}$ and $\textbf{y}$ defined in open sets (this is important: the set $(0,2 \pi]\times \mathbb{R}$ is not open, and a parametrization has to be defined into an open set).

Do someone know an atlas with only one chart? For example, with a reparametrization. The only thing I know is that the atlas with one chart exists in this case$^*$.

*Introduction to Topological Quantum Matter & Quantum Computation, Tudor D. Stanescu

PD: Something similar in General Relativistic to changing Schwarzchild coordinates by Kruskal–Szekeres ones to extend the well-behaved domain.

Best Answer

You can use the chart $$ \mathbf x(u,v) = \left(\frac{u}{\sqrt{u^2+v^2}}, \frac{v}{\sqrt{u^2+v^2}}, \log \sqrt{u^2+v^2} \right)\,,$$ defined on $\mathbf R^2 \setminus \{(0,0)\}$. The formula is simpler in polar coordinates, $$ \mathbf x(r,\theta) = \left( \cos \theta, \sin \theta, \log r\right)\,. $$

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