Explicit equation for geodesic mapping

Question

Let $M$ and $N$ be two Riemannian manifolds. A geodesic mapping is a diffeomorphism $F: M \to N$. $F$ maps each geodesic arc on $M$ to a geodesic arc on $N$. I am looking for the explicit equation satisfied by $F$. For each curve $\gamma$ on $M$, $F \circ \gamma$ is a geodesic curve on $N$, that is: for all $\gamma$ such that $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$, one has $\nabla_{\dot{(F \circ \gamma})} \dot{(F \circ \gamma)} = 0$ which will be expanded out to obtain an equation of $F$ that I am interested in. So far, I have been unable to expand this out. I think the question is quite natural, but I haven't found a reference yet.