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 Ã¢Â†Â’ 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 (M, g) and (N, h) 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 (M, g) and (N, h) 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 (M, g) is the unit sphere Sⁿ with its usual round metric and (N, h) 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 Ã¢Â†Â’ Rⁿ⁺¹ given by φ(x) = 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 (D, g) be the unit disc D Ã¢ÂŠÂ‚ R² equipped with the Euclidean metric, and let (D, h) 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 Ã¢Â†Â’ 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 Ã¢Â†Â’ D is a geodesic map, because hyperbolic geodesics in the Klein model are (Euclidean) straight line segments.

References

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