It is well known that every $C^1$ manifold admits a smooth manifold structure. What if we relax the definition of smooth manifold so the transition maps need only be differentiable? Does every such "differentiable manifold" admit a compatible $C^1$ atlas?
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$\begingroup$ $C^\infty \subset C^1$. A $C^\infty$ atlas is also a $C^1$ atlas, $\endgroup$– Paul SinclairCommented Aug 30, 2023 at 15:54
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$\begingroup$ @PaulSinclair I am aware of that. How is that related to my question? $\endgroup$– Carla_Commented Aug 30, 2023 at 20:52
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$\begingroup$ I misunderstood your question. $\endgroup$– Paul SinclairCommented Aug 30, 2023 at 21:09
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$\begingroup$ Do you mean a structure, that is not necessary continuously differentiable, but at least differentiable? $\endgroup$– EricCommented Aug 31, 2023 at 16:20
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$\begingroup$ @Eric Yes that is what i mean. $\endgroup$– Carla_Commented Aug 31, 2023 at 16:27
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