Example of $C^k$ manifold but not $C^{k+1}$ manifold

Asked Oct 23 '20 at 04:08

Active Oct 23 '20 at 04:08

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I try to construct an example which is not smooth but is $C^k$ manifold, but fail.

In past, I always think (without any think) the graph of $y=|x|$ is not $C^1$ manifold, but just consider $(x, |x|) \rightarrow x$, I think the graph of $y=|x|$ has smooth differential structure.

Therefore, seemly, there usual examples always have smooth differential structure. So, I want to know an example which is $C^k$ manifold but not $C^{k+1}$ manifold.

Simpler is better. Very Thanks.

asked Oct 23 '20 at 04:08

Enhao Lan

5,829

See the existence and uniqueness theorems of https://en.m.wikipedia.org/wiki/Differential_structure. – Dustan Levenstein Oct 23 '20 at 04:43
It is a theorem that every $C^k$ manifold has a compatible $C^\infty$ structure. – abhi01nat Oct 23 '20 at 04:45

Example of $C^k$ manifold but not $C^{k+1}$ manifold

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