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We call an atlas smooth if each of its transition maps (or change of charts) is smooth (or $C^\infty$). A differential structure is a maximal smooth atlas. A smooth manifold is a topological manifold together with a smooth structure.

These are the definitions of my lecture notes. I am also wondering now why we have not defined what a smooth structure is and if one has to replace smooth structure with differentiable structure instead. In Loring Tu's An Introduction to Manifolds it is also mentioned that it has to be a differentiable structure.

But I tried to compare both definitions on Wikipedia and I can not really tell the crucial difference between these both terms. Both are somehow dealing with maximal atlases.

Could you give me an explanation about the right definition about smooth manifolds? Thank you.

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    $\begingroup$ I think the main point here is what differentiable means. Tipically it is a synonym of smooth, which means $C^\infty$. But you can get any $C^r$ -differentiable structure by setting the transition maps to be $C^r$ functions instead of $C^\infty$ (even $r=0$). $\endgroup$ – Dog_69 Dec 16 '17 at 1:04
  • $\begingroup$ A chart $\phi_i$ makes the manifold locally isomorphic to an open of $\mathbb{R}^n$, but it can happen $\phi_j^{-1}\phi_i$ is only continuous, or $C^1$, or $C^k$, in that case it doesn't make sense to talk of $C^{k+1}$ functions from the manifold to $\mathbb{R}$ $\endgroup$ – reuns Dec 16 '17 at 2:00
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"Smooth structure" and "differentiable structure" are synonyms (in the context of your notes), so you're not missing anything.

(Note that there do exist variations on the definition where you require the transition maps to be $C^k$ for some fixed $k<\infty$ instead of $C^\infty$. A maximal atlas of such charts could be called a "$C^k$ structure" or "$C^k$ differentiable structure", or in contexts where $k$ is clear perhaps just "differentiable structure". In contrast, "smooth structure" pretty much always means the $C^\infty$ version.)

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