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?
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4$\begingroup$ 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 GerigSep 3, 2013 at 2:50
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1$\begingroup$ (This construction clearly shows it is not unique -- add random bump functions) $\endgroup$– Chris GerigSep 3, 2013 at 2:54
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$\begingroup$ Where "constant vector field" means constant in some coordinate system on the ball, of course. $\endgroup$– Anthony CarapetisSep 3, 2013 at 3:20
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1 Answer
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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.
