Why isn't the source diffemorphic to Euclidean space but to a cylinder?

@MaryStar The circle $S^1$ is not diffeomorphic to $\mathbb{R}$, but it contains closed loops which cannot be deformed smoothly into a single point. Since it enters a factor of $\mathbb{R}\times S^1$ the product cannot be diffeomorphic to $\mathbb{R}^2$ either. Strictly speaking you don't need to know that for your task, but you have to write down the domain of definition for your charts. Even if you don't know that $\mathbb{R}\times S^1$ is not diffeomorphic to Euclidean space it is not clear that it is suitable as domain of definition for a chart. So you should try to find something else.

I got stuck right now... Could you explain to me how we search for an atlas?

@MaryStar You have to look for coordinate patches which are homeomorphic to Euclidean space (e.g. Euclidean space itself, a unit disc, a product of intervals, ...). In the present case I already provided a possible choice, and the maps are also already given.

Why do we cover the $S^1$ in the domain of definition by these two line segments?

@MaryStar Because (each of) these line segments is/are diffeomorphic to an interval, while $S^1$ is not.

Ahaa... Ok... $$$$ Consider we have the half-cone $$S: \ x^2+y^2=z^2, z>0$$ It can be parametrized by $$\sigma (u,v)=(u\cos v, u\sin v, u)$$ Do we take a subset of the domain of definition so that it is homeomorphic to a unit disc, to find the atlas in this case?

@MaryStar Two subsets will do, one won't suffice in this case. You can also use more than two (as many as you want, actually), you only have to cover the manifold with the images of the maps, which should be defined on something diffeormorphic to Euclidean space (this is what is usually used in the definitions).

Which is the original domain of definition of the half-cone? $[0,2\pi]\times [0,2\pi]$ ?

@MaryStar You have written down the parametriziation so you should decide, there is no general rule. I'd take $[0,2\pi)\times (0,\infty)$ (which is not nice since it has a boundary) or $(0,2\pi)\times (0,\infty)$ allowing the lack of one ray. If you take $(0,2\pi)\times (0,\infty)$ and $(-\pi,\pi)\times (0,\infty), $ say, you'd get your atlas.

I see... Thank you so much for your help!! :-)

Could I ask you also an other question?

This is hard to read here in the comments section. The set $f(\mathcal{O})$ with your choices is the cone minus a ray on the cone which is perpendicular to the circles which make up the cone. The map $f$ is not a diffeomorphism since it covers the cone several times.

Ah ok... And which is the equation of that ray? Is this ray on the $z$-axis? @Thomas