Let $M^n$ be a riemannian manifold and $p_0$ a point in $M$. Let $U$ be a normal neighbourhood of $p_0$, image of $B_{\delta}(0) = \{ x \in T_{p_0} M : \left\lvert x \right\rvert < \delta \}$ under the exponential map $exp_{p_0}$. For a fixed non zero vector $v \in B_{\delta}(0)$, define a vector field $V = V_v$ on $U$ by the following rule: given $p \in U$, $V(p)$ is the parallel transport of $v$ along the radial geodesic starting at $p_0$ and ending at $p$.
My question is: can we obtain a formula for $\nabla_X V$, where $X$ is a vector field defined on $U$ (and $\nabla$ is the Levi-Civita connection of $M$)?