- Geodesic map
In

mathematics — specifically, indifferential geometry — a**geodesic map**(or**geodesic mapping**or**geodesic diffeomorphism**) is a function that "preservesgeodesic s". More precisely, given two (pesudo-)Riemannian manifold s ("M", "g") and ("N", "h"), a function "φ" : "M" → "N" is said to be a geodesic map if

* "φ" is adiffeomorphism 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 theinverse 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"-

dimension alEuclidean space **E**^{"n"}with its usual flat metric, then any Euclideanisometry 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 ofradius 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" ⊂**R**^{2}equipped with the Euclidean metric, and let ("D", "h") be the same disc equipped with a hyperbolic metric (as in thePoincaré disc model of hyperbolic geometry). Then, although the two structures are diffeomorphic via theidentity map "i" : "D" → "D", "i" is "not" a geodesic map, since "g"-geodesics are always straight lines in**R**^{2}, whereas "h"-geodesics can be curved.**References*** 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**External links***

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2010.*

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