Let $ M$ a complete riemannian manifold, if $ x, y \in M $ such that $d (x, y) <\min \{i_ {M} (x), i _{M} (y) \} $, $ l = d (x, y) $, there is a unique minimizing geodesic $ \gamma: [0, l] \to M $ by connecting the points $ x $ and $ y $. Now consider $\Gamma $ a geodesic variation, such that $\Gamma(s,0)=\gamma(t)$ which is associated with the unique Jacobi field $ X $, so that $X(0)=v$ and $X(l)=w$, where $w=L_{xy}v$ (parallel transport o vector $v$). My question is if I can find a coordinate system where I can write the vector field $X$ in terms of parallel fields.
Edit: $i_{M}(x),i_{M}(y)$ is the injectivity radius of $x$,$y$ respectively.