- Geodesic map
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
diffeomorphismof "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 spaceE"n" with its usual flat metric, then any Euclidean isometryis 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
radius2 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 modelof 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 MathSciNet|id=679133
* 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 MathSciNet|id=1118149
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