2
$\begingroup$

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 $a_k=b$.

Edit

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

$\endgroup$
  • $\begingroup$ 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
3
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.