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?
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?
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.