Suppose $M$ be a manifold and $v, w$ be (smooth) sections of vector bundle $E \to M$ along a curve $\gamma, \gamma':[0,1] \to M$.
Assume that $\gamma$ and $\gamma'$ don't meet excepting at their initial point(t = 0) and each of them don't intersect theirselves. In addition, $v(0) = w(0)$.
Then, does there exist an extension of $s \in \Gamma(E)$ satisfies $s \circ \gamma = v$ and $s \circ \gamma' = w$?
I have an idea using local spherical coordinates and bump function, but maybe there exist another method to do this. Any suggestions is appreciated.
ADD
We need additional assumption for making sense. $\{ \dot{\gamma}(0),\dot{\gamma}'(0)\}$ is linearly independent in $T_{\gamma(0)} M$.