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?
2026-04-08 01:49:08.1775612948
A $C^k$ manifolds ($k>1$) has a $C^\infty$ atlas: reference request
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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.