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Let $M$ be a smooth manifold and $\varphi=(x^1,...,x^n)$ local coordinates defined on $U$. Then the coordinate vector fields $$(\varphi^{-1})_*\frac{\partial}{\partial x^i}$$ determines smooth vector fields on $U$. Can these always be smoothly extended to $M$? (It may vanish outside $U$.) I think they should, in order to make other definitions like the connection and curvature tensor valid. I consider the books of John M. Lee.

I know that a general smooth vector field on an open set of $M$ cannot always be smoothly extended, take for example: $X:(0,1)\to T\mathbb R, x\to \frac 1{x(x-1)}\frac{\partial}{\partial x^1}$.

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    $\begingroup$ Well, you can make them vanish smoothly, and then they are extended to the whole $M$. If you want to extend them smoothly without make them vanish, then it's not always possible. Think of $M=S^2$. $\endgroup$ – Jackozee Hakkiuz Jul 10 '18 at 18:52
  • $\begingroup$ @JackozeeHakkiuz, how can you make them vanish smoothly? Via partition of unity? Do you maybe have a reference? $\endgroup$ – Mark Jul 10 '18 at 19:03
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    $\begingroup$ I have tried to ask the same thing here. The answer is that you can't always extend a function defined on an open subset to a larger set. The best i can do is to extend the restriction of the function to some closed subset inside the original domain. $\endgroup$ – Sou Jul 21 '18 at 14:20

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