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Let $M=\{(t,f(t))\mid t\in (-1,1)\}$ a sub-manifold of $\mathbb R^2$ of dimension $1$ (so $f$ is at least $\mathcal C^1$). A theorem of my course says that for all $a\in M$, there is a chart $(\varphi ,W)$ where $W$ is an open of $\mathbb R^2$ s.t. $\varphi (a)=0$ and $$\varphi (M\cap W)=\{(x,y)\in W\mid y=0\}.$$

What could be such a Chart in my case ? I was thinking about something as using the derivative, but I'm not sure it really make sense... I'm quite confuse with those charts... can someone explain ?

user659895
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  • What about $W=\mathbb R^2$ and $\varphi=\pi_1$, the projection onto the first factor? – Dog_69 May 23 '19 at 09:19
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    I think you should replace $I$ by an open interval. As you define it, it is a manifold with boundary which cannot be a submanifild of $\mathbb R^2$. Are manifolds topological manifolds or $C^k$-manifolds for $k \ge 1$? – Paul Frost May 23 '19 at 13:01
  • @PaulFrost Good point. I didn't notice $I$ was closed. – Dog_69 May 23 '19 at 14:55
  • @Dog_69: I corrected it. I'm quite surprise that the projection is a diffeomorphism... but it looks to be one, indeed. – user659895 May 23 '19 at 18:04

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