Geodesic mapping

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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.