I remember that there is a theorem saying that "any $C^k (k>1)$ manifold has a smooth atlas" so that, instead of consdering $C^k$ manifolds, we study smooth manifolds.
What is this theorem called and where can I find its proof?
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1$\begingroup$ Does every theorem needs to have name? $\endgroup$– CreatorAug 16, 2016 at 22:24
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3$\begingroup$ @Creator Definitely not, but I'm having trouble searching for its proof. That's why I asked whether there is a name for this theorem. Actually, I'm not sure whether the statement is correct. I just remember I heard that from someone someday, but I have no idea who it was and what exactly the statement was. $\endgroup$– RubertosAug 16, 2016 at 22:37
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$\begingroup$ I think Hirsch has this. $\endgroup$– Aloizio Macedo ♦Aug 17, 2016 at 0:46
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$\begingroup$ For a textbook reference (Hirsch), see here. $\endgroup$– Moishe Kohan on strikeNov 11, 2021 at 15:21
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1 Answer
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It's a result by Hassler Whitney, in one of his earlier papers on manifolds, that any maximal $C^r$ atlas, for $r>0$, contains a $C^\infty$ atlas. I can't find the specific paper at the moment, but it should be in volume I of his collected publications. My guess would be either [24] or [28] in the bibliography.
Most textbooks I know of, such as Jeffrey Lee's Manifolds and Differential Geometry, mention the result, but don't find it worth proving, so I would suggest looking for Whitney's original paper if you want to see the proof.
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1$\begingroup$ It's reference [28]. See the paper on jstor here: jstor.org/stable/1968482?seq=1#metadata_info_tab_contents. $\endgroup$– user141854Feb 13, 2019 at 18:28