I am looking at the following exercise:
The surface obtained by rotating the curve $x = \cosh z$ in the $xz$-plane around the $z$-axis is called a catenoid. Describe an atlas for this surface.
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I have done the following:
Let $\gamma (u)=(\cosh u, 0, u)$ be a parametrization of the profile curve $x = \cosh z$.
A point of the catenoid has as vector $$\sigma (u,v)=(\cosh u\cos v, \cosh u\sin v, u)$$
Is this correct so far?
How can we find the atlas?
You defined a map from $\mathbb{R}\times S^1$ onto the catenoid. As a parametrization this is fine.
When searching for an atlas then the problem with this is that the source is not diffemorphic to Euclidean space but to a cylinder. Just cover the $S^1$ in your domain of definition by two line segments $I_1, I_2$ of length $\pi+\varepsilon$, say, and restrict your map to the resulting sets $I_k\times \mathbb{R}$. This will give you an atlas consisting of two charts.