maximal smooth atlas

An atlas is a collection of charts that cover the entire topological manifold. A smooth atlas includes a condition that the overlap of charts be compatible with calculus how we know it in $\mathbb{R}$.

Definition

A maximal smooth atlas for the manifold $(M,n)$ is $\mathcal{A}=\{{\phi }_i:U_i\to V_i{U}_i\subset M,V_i\subset {\mathbb{R}}^n\text{open}{\}}_{i\in I}$ such that

1. $\cup U_i=M$
2. $\forall i,j\in I$, the map ${\phi }_i\circ {\phi }_j^{-1}:{\phi }_i(U_i\cap U_j)\to {\phi }_j(U_i\cap U_j)$ is smooth. Note we only care about the intersections. If something lives outside the intersections, i.e. it is only in the “middle” of a chart, then the calculus already works well because it is homeomorphic to ${\mathbb{R}}^n$ inside this openset. In other words, we want calculus to work nice as we move between charts.
3. Maximality - nothing is left out. Given a homeomorphism ${\phi }^{\prime }:U^{\prime }\to V^{\prime }$ between open sets in $M$ and $\mathbb{R}$, such that \begin{align}&\forall \varphi \in \mathcal{A} \\ \varphi^\prime \circ \varphi^{-1}&: \varphi(U \cap U^\prime) \to \varphi^\prime(U \cap U^\prime) \\ &\text{and}\\ \varphi \circ (\varphi^\prime)^{-1}&: \varphi^\prime(U \cap U^\prime) \to \varphi(U \cap U^\prime) \\ \end{align} $$ are smooth maps, then $\varphi^\prime \in \mathcal{A}$. Said another way, if we happen to find some other smooth chart that works with the smooth manifold structure, then we can assume that it was already in $\mathcal{A}$.

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References