Let $M$ be a smooth manifold and $\gamma:I\to M$ a smooth curve on $M$. How can I construct a local chart $(U,\psi)$ such that $U\cap \gamma(I)\ne \emptyset$ and $\gamma(t)\equiv(\gamma^1(t),0,\ldots,0)$ for all $t$?
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Do you have to construct a local chart for each point in $\gamma(I)$? – edm Aug 31 '17 at 15:16
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2Where $\gamma'(t)\ne 0$, use the constant rank theorem. At points where $\gamma'(t)=0$, you can't. – Jack Lee Aug 31 '17 at 22:11