# Geodesic mapping

## Definition

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

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