Geodesic map

In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ : M → N is said to be a geodesic map if

• φ is a diffeomorphism of M onto N; and
• the image under φ of any geodesic arc in M is a geodesic arc in N; and
• the image under the inverse function φ⁻¹ of any geodesic arc in N is a geodesic arc in M.