Let $M$ be a smooth manifold of dimension $n$. Suppose $X \in \mathfrak{X}(M)$ and $p \in M$. Then, we may find a curve $\gamma:(-\epsilon, \epsilon) \rightarrow M$ where $\gamma(0) = p$ and $\gamma'(0) = X(p)$.
I want the other way around: suppose I have a curve $\gamma:(-\epsilon, \epsilon) \rightarrow M$. From this curve $\gamma$, is it possible to construct $X \in \mathfrak{X}(M)$ such that $X(\gamma(t)) = \gamma'(t)$ for all $t \in (-\epsilon, \epsilon)$? If this is impossible to make a global construction, can I make a local construction?
Thanks in advance.
