# Smooth curve segments and smooth charts

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 $$a=a_0 such that $$\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 $$a=a_0 such that $$\gamma([a_{i-1},a_i])$$ is contained in $$U_i$$ for each $$i=1,\dots,k$$.

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

Edit

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

• 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 $b$. – Paul Frost Mar 2 at 17:30
• Thank-you :) Can you show me how can i prove my statement? – 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.