For each nonnegative integer $n$, the Euclidean space $\mathbb{R}^n$ is a smooth $n$-manifold with the smooth structure determined by the atlas $\mathcal{A}=(\mathbb{R}^n,\mathbb{1}_{\mathbb{R}^n})$.
Any chart $(U,\varphi)$ contained in the given maximal smooth atlas is called smooth chart or smooth coordinate chart.
Since $\mathcal{A}$ is a smooth atlas, is contained in a unique maixmal smooth atlas, called smooth structure determined by $\mathcal{A}.$
Who are the smooth coordinate chart $(U,\varphi)$ for $\mathbb{R}^n$ respect to this smooth smooth structure?
It has to happen that $\varphi\circ\mathbb{1}_{\mathbb{R}^n}^{-1}\colon\mathbb{1}_{\mathbb{R}^n}(\mathbb{R}^n\cap U)\to \varphi(\mathbb{R}^n\cap U)$ is $C^{\infty}$, that is $\varphi\colon U\to \hat{U}:=\varphi(U)$ is $C^{\infty}$, same reasoning for $\mathbb{1}_{\mathbb{R}^n}\circ\varphi^{-1}$. Therefore, with respect to this smooth structure, the smooth coordinate charts for $\mathbb{R}^n$ are exactly those charts $(U,\varphi)$ such that $\varphi$ is a diffeomorphism.
So, in the case of $\mathbb{R}^n$, then we can describe the maximal atlas, right?
Question. Are there other cases of manifolds in which we can fully describe some maximal atlas?
Thanks!