Geodesic map

In mathematics—specifically, in differential geometry—a geodesic map is a function that "preserves geodesics". More precisely, given two Riemannian manifolds and, 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.

Examples

If and are both the n-dimensional Euclidean space Eⁿ with its usual flat metric, then any Euclidean isometry is a geodesic map of Eⁿ onto itself.
Similarly, if and are both the n-dimensional unit sphere Sⁿ with its usual round metric, then any isometry of the sphere is a geodesic map of Sⁿ onto itself.
If is the unit sphere Sⁿ with its usual round metric and is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space Rⁿ⁺¹, then the "expansion" map φ : Rⁿ⁺¹ → Rⁿ⁺¹ given by φ = 2x induces a geodesic map of M onto N.
There is no geodesic map from the Euclidean space Eⁿ onto the unit sphere Sⁿ, since they are not homeomorphic, let alone diffeomorphic.
The gnomonic projection of the hemisphere to the plane is a geodesic map as it takes great circles to lines and its inverse takes lines to great circles.
Let be the unit disc D ⊂ R² equipped with the Euclidean metric, and let be the same disc equipped with a hyperbolic metric as in the Poincaré disc model of hyperbolic geometry. Then, although the two structures are diffeomorphic via the identity map i : D → D, i is not a geodesic map, since g-geodesics are always straight lines in R², whereas h-geodesics can be curved.
On the other hand, when the hyperbolic metric on D is given by the Klein model, the identity i : D → D ''is'' a geodesic map, because hyperbolic geodesics in the Klein model are straight line segments.