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Let $A_S$ be the maximal atlas of an embedded submanifold $S$, Let $A_M$ be the maximal atlas of the ambient space $M$. Let $A_M'$ be the set of charts in $A_M$ with their domains restricted on $S$. Is $A_S\subset A_M'$? Why?

Intuitively, every chart $\phi$ of $S$ can be extended to a chart $\phi'$ of $M$ because S inherits the topology of $M$. And even if there is an extension, can we make the extension $\phi'$ compatible with $A_M$? Further, if we work on smooth manifold, is it possible to make $\phi'$, if any, compatible with the smooth structure?

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  • $\begingroup$ How exactly are elements of $A_M'$ "charts" if $\dim S<\dim M$? They will be maps between subsets of $S$ and certain non-open subsets of $\mathbb{R}^{\dim M}$, rather than maps between subsets of $S$ and open subsets of $\mathbb{R}^{\dim S}$. $\endgroup$ – Eric Wofsey Oct 20 '16 at 23:58
  • $\begingroup$ I guess you probably mean to think of $\mathbb{R}^{\dim S}$ as a subset of $\mathbb{R}^{\dim M}$, so that it is meaningful to ask whether a chart in $A_S$ is "the same" as an element of $A_M'$. $\endgroup$ – Eric Wofsey Oct 21 '16 at 0:01
  • $\begingroup$ @EricWofsey From the definition of embedded submanifold, there is a "submanifold chart "$(\phi, U) \in A_M$ for each point $p\in S$, s.t. $\phi(U\bigcap S)= \phi(U)\bigcap (\mathbb R^k)$, k is dimension of S. And the restriction of $\phi$ on $S$ is a chart for $S$. $A_M'$ contains all the charts of $S$ obtained in this way. I want to know if this exhausts the compatible charts for $S$. $\endgroup$ – user136592 Oct 21 '16 at 0:04

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