Let $M$ be a smooth Riemannian manifold with Levi-Civita connection $\nabla$. Let $p \in M$ and $X$ be a smooth vector field along a smooth curve $c : (-\varepsilon, \varepsilon) \to M$ with $c(0) = p$ and $v = c'(0) \neq 0$. I believe that the following is true:
For any vector $w \in T_p M$ which is not colinear to $v$ there exists a smooth extension $X_w$ of $X$ to the geodesic ball $B_\delta(p)$ such that $X_w$ is parallel along the geodesic $t \mapsto exp(tw)$ on $B_\delta(p)$.
My thought is that $\delta$ has to be small enough so that $c$ and the geodesic $t \mapsto exp(tw)$ do not intersect each other (except at $p$). Does this $\delta$ exist?
Also, how one "glues" the parallel transport of $w$ along $t \mapsto exp(tw)$ and $X$ in order to obtain a smooth vector field on the ball? I assume the hypotheses that $w$ and $v$ are not colinear comes into play here, but I don't know how to prove such a smooth extension exists. Any help?