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?