Let $M$ be a smooth manifold with or without boundary, and let $\gamma:[a,b]\to M$ be a smooth curve.

I want to show that there exist a finite partition $a=a_0<a_1<\dots<a_k=b$ such that $\gamma([a_{i-1},a_i])$ is contained in the domain of a single smooth chart for each $i=1,\dots,k$.

Here is my argument but I'm not able to conclude it

By compactness of $\gamma([a,b])$ there exists a finite number of smooth charts $(U_i,\varphi_i)_{i=1}^k$ such that $$\gamma([a,b])\subseteq U_1\cup\dots\cup U_k.$$

Suppose $\gamma(a)\in U_1$, so we have $a\in \gamma^{-1}(U_1)$ wich is open in $[a,b]$. So there is an interval of the form $[a,\varepsilon)$ contained in $\gamma^{-1}(U_1)$ with $\varepsilon>a$. Let $\delta:=$sup$\{\varepsilon>a:[a,\varepsilon)\subseteq \gamma^{-1}(U_1)\}$. Then I can show that $[a,\delta) \subseteq \gamma^{-1}(U_1)$. Thus we have $\gamma([a,\delta))\subseteq U_1$ and $\delta>a$.

Now suppose that $\gamma(\delta)\in U_2$, so $\delta\in \gamma^{-1}(U_2)$ which is open in $[a,b]$. Thus there is $\varepsilon>0$ such that $(\delta-\varepsilon,\delta+\varepsilon)\subseteq \gamma^{-1}(U_2)$ and $\delta-\varepsilon>a$. Let $a_1=\delta-\varepsilon/2$. Thus we have $\gamma([a,a_1])\subseteq U_1$ and $\gamma(a_1)\in U_2$.

Repeating the argument with $a_1$ in place of $a$, then i can find $a_2$ such that $\gamma([a_1,a_2])\subseteq U_2$ and $\gamma(a_2)\in U_3$. Contiuing this way I find points $a=a_0<a_1<\dots<a_k$ such that $\gamma([a_{i-1},a_i])$ is contained in $U_i$ for each $i=1,\dots,k$.

I'm not able to show that $a_k=b$.


As noted in the comment, this argument is wrong. So how can I prove this statement?

  • $\begingroup$ You argument does not work. It is possible that $\gamma(a_2)$ is again in $U_1$ and in no other $U_i$. Thus you cannot be sure that your process stops at $b$. $\endgroup$ – Paul Frost Mar 2 at 17:30
  • $\begingroup$ Thank-you :) Can you show me how can i prove my statement? $\endgroup$ – Minato Mar 2 at 17:56

Let $\{ U_\alpha \}_{\alpha \in A}$ be any open cover of $\gamma([a,b])$ where the $U_\alpha$ are the domains of charts. Then the $V_\alpha = \gamma^{-1}(U_\alpha)$ form an open cover of $[a,b]$. You can now take a Lebesgue number for $\mathfrak{V} = \{ V_\alpha \}_{\alpha \in A}$ to find the desired partition. This is a number $r > 0$ such that each set of diameter $< r$ is contained in some $V_\alpha$. See https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma or any textbook on general topology. Also see my answer to Relation between Riemann sums and oscillation of a bounded function. The proof can easily be adapted to general covers.

  • $\begingroup$ Thank-you very much! :) $\endgroup$ – Minato Mar 3 at 17:18

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