Given a Riemannian manifold $M$, and a smooth curve $\gamma: I \to M$. We know how to define the parallel transport along the curve, denoted by $\tau_t$. Now at point $T_{\gamma(0)}M$ we can take a basis $v_1, \ldots, v_n$, we have $\tau_t(v_1), \ldots, \tau_t(v_n)$. Those form a basis at $T_{\gamma(t)}M$, because parallel transport is a linear isomorphism.
Now let $W$ be a vector field on $M$, $W_\gamma(t)= \sum c_i(t)\tau_t(v_i)$. Is it true that $c_i(t)$ must be smooth?