A smooth vector field along a curve $c:[0,1] \to M$, such that $V(0) = 0$, $V(1) = w$ for some fixed $w$.

Question

A smooth vector field along a curve $c:[0,1] \to M$ (where M is a smooth manifold), such that $V(0) = 0$, $V(1) = w$ for some fixed $w$. Is it always possible to construct such a vector field along the curve $c$?

Answer 1

Yes. The pullback bundle $c^{*}(TM)$ is trivial (being a bundle over a contractible space). Choose some trivialization $\Phi \colon c^{*}(TM) \rightarrow [0,1] \times \mathbb{R}^n$ (where $\dim M = n$). Given $w \in T_{c(1)}M$, set $\tilde{w} := \pi_2(\Phi(1,w))$ and let $f \colon [0,1] \rightarrow \mathbb{R}^n$ be any smooth curve with $f(0) = 0$ and $f(1) = \tilde{w}$. Then $V(t) := \Phi^{-1}(t,f(t))$ is a smooth vector field along $c$ with $V(0) = 0$ and $V(1) = w$.