Geodesic map

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 (pesudo-)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".


* If ("M", "g") and ("N", "h") are both the "n"-dimensional Euclidean space E"n" with its usual flat metric, then any Euclidean isometry is a geodesic map of E"n" onto itself.

* Similarly, if ("M", "g") and ("N", "h") are both the "n"-dimensional unit sphere S"n" with its usual round metric, then any isometry of the sphere is a geodesic map of S"n" onto itself.

* If ("M", "g") is the unit sphere S"n" 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 R"n"+1, then the "expansion" map "φ" : R"n"+1 → R"n"+1 given by "φ"("x") = 2"x" induces a geodesic map of "M" onto "N".

* There is no geodesic map from the Euclidean space E"n" onto the unit sphere S"n", since they are not homeomorphic, let alone diffeomorphic.

* 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.


* cite book
last = Ambartzumian
first = R. V.
title = Combinatorial integral geometry
series = Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics
publisher = John Wiley & Sons Inc.
location = New York
year = 1982
pages = pp. xvii+221
isbn = 0-471-27977-3
* cite book
last = Kreyszig
first = Erwin
title = Differential geometry
publisher = Dover Publications Inc.
location = New York
year = 1991
pages = pp. xiv+352
isbn = 0-486-66721-9

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