# Can a tangent vector extend to a vector field?

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.

• 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. – Chris Gerig Sep 3 '13 at 2:50
• (This construction clearly shows it is not unique -- add random bump functions) – Chris Gerig Sep 3 '13 at 2:54
• Where "constant vector field" means constant in some coordinate system on the ball, of course. – Anthony Carapetis Sep 3 '13 at 3:20

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.