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.

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 → 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 ⊂ 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 → 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 (Euclidean) straight line segments.

References

  • Ambartzumian, R. V. (1982). Combinatorial integral geometry. Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. New York: John Wiley & Sons Inc. pp. xvii+221. ISBN 0-471-27977-3. MR 0679133.
  • Kreyszig, Erwin (1991). Differential geometry. New York: Dover Publications Inc. pp. xiv+352. ISBN 0-486-66721-9. MR 1118149.