Geodesic mapping

Definition

Let $M$ and $N$ be Riemannian manifolds. A geodesic mapping from $M$ to $N$ is a diffeomorphism from $M$ to $N$ that sends geodesics to geodesics, such that its inverse also sends geodesics to geodesics.

Another way of saying this is that if $M$ is a differential manifold, then two Riemannian metrics $g_{1}$ and $g_{2}$ of $M$ are related by a geodesic mapping if the geodesics for $g_{1}$ are precisely the same as the geodesics for $g_{2}$.