Smooth atlas

Definition

Let $M$ be an $n$-dimensional topological manifold. A smooth atlas on $M$ is a collection of charts $\mathcal A=\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in I}$ such that

1. $\{U_\alpha\}_{\alpha\in I}$ covers $M$, and
2. for all $\alpha,\beta$ with $U_\alpha\cap U_\beta\neq\varnothing$, the transition map $\varphi_\beta\circ\varphi_\alpha^{-1}:\varphi_\alpha(U_\alpha\cap U_\beta)\to \varphi_\beta(U_\alpha\cap U_\beta)$ is a smooth map between open subsets of $\mathbb{R}^n$.

Two atlases are compatible if their union is again a smooth atlas; compatibility is an equivalence relation. The maximal atlas generated by $\mathcal A$ is the set of all charts smoothly compatible with every chart in $\mathcal A$. Choosing a maximal smooth atlas is exactly what turns $M$ into a smooth manifold .

Examples

1. On $\mathbb{R}^n$, the single chart $(\mathbb{R}^n,\mathrm{id})$ is a smooth atlas; its generated maximal atlas is the standard smooth structure on $\mathbb{R}^n$.
2. On $S^n$, the two stereographic projection charts form a smooth atlas because their overlap transition map is smooth; the maximal atlas they generate is the usual smooth structure on the sphere.
3. If $M$ and $N$ are smooth manifolds with atlases $\mathcal A_M$ and $\mathcal A_N$, then the product charts $(U\times V,\varphi\times\psi)$ with $(U,\varphi)\in\mathcal A_M$ and $(V,\psi)\in\mathcal A_N$ form a smooth atlas on $M\times N$, characterized by the fact that the projections are smooth maps .