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I want to know another structure in S1. I want that it not be diffeomorphic to the usual structure.

Thanks!

hal97
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  • Transport the usual structure on $\mathbb{R}$ through a noncontinuous bijection $\mathbb{R} \rightarrow S^1$. – Aphelli Feb 14 '19 at 19:30
  • By a bijection that it not be diffeomorpism. For example if I have the atlas $\varphi (t) = e^{2\pi i t} $ and I do $\psi = f \circ \varphi $ where $f (t) = (t -1/2)^3$, results that $\psi \varphi^{-1} $ its not a diff. – hal97 Feb 15 '19 at 12:29

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I assume you want the topology to stay the same. If so, there is no other differentiable structure on the circle. See the classification of 1-dimensional manifolds. The circle will be 1-dimensional as a smooth manifold no matter what, just because of the topology. And the only one in the list of 1-dimensional manifolds which is even homeomorphic to the circle is the standard smooth structure on the circle, so it must be that one.

If you're interested in things like this, though, maybe look into Milnor's exotic spheres.

user555203
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  • Great! but... this is not my question... But your answer is more interesting, and Im going to investigate about it – hal97 Feb 15 '19 at 12:33
  • @hal97 I think cs47511's answer completely answers your question: any two smooth structures on $S^1$ must be diffeomorphic. – Rachid Atmai Nov 22 '20 at 20:21