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Let $M$ be a smooth manifold and $p\in M$. I would like to know whether any tangent vector $X_p \in T_pM$ extends to a vector field over $M$.

If so is it unique? How can I construct it?

Thank you.

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    $\begingroup$ Take a bump function, with support in a small ball centered on your point, and multiply it by the constant vector field (value $X_p$)... this has fixed points outside of your ball. $\endgroup$ – Chris Gerig Sep 3 '13 at 2:50
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    $\begingroup$ (This construction clearly shows it is not unique -- add random bump functions) $\endgroup$ – Chris Gerig Sep 3 '13 at 2:54
  • $\begingroup$ Where "constant vector field" means constant in some coordinate system on the ball, of course. $\endgroup$ – Anthony Carapetis Sep 3 '13 at 3:20
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First notice that this problem is easy when you're in $\mathbb{R}^n$. Then use a small coordinate patch and a bump function to extend the vector field as you wish.

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