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In [[mathematics]]—specifically, in [[differential geometry]]—a '''geodesic map''' (or '''geodesic mapping''' or '''geodesic diffeomorphism''') is a [[Function (mathematics)|function]] that "preserves [[geodesic]]s". More precisely, given two ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]]s (''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''-[[dimension]]al [[Euclidean space]] '''E'''''n'' with its usual flat [[Riemannian metric|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 [[hypersphere|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 [[homeomorphism|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'' ⊂ '''R'''2 equipped with the Euclidean metric, and let (''D'', ''h'') be the same disc equipped with a [[hyperbolic geometry|hyperbolic]] metric as in the [[Poincaré disc model]] of hyperbolic geometry. Then, although the two structures are diffeomorphic via the [[identity function|identity 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.
* On the other hand, when the hyperbolic metric on ''D'' is given by the [[Beltrami-Klein model|Klein model]], the identity ''i'' : ''D'' → ''D'' ''is'' a geodesic map, because hyperbolic geodesics in the Klein model are (Euclidean) straight line segments.

==References==

* {{cite book
| last = Ambartzumian
| first = R. V. | authorlink = Rouben V. Ambartzumian
| 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 = xvii+221
| isbn = 0-471-27977-3
| mr=679133}}
* {{cite book
| last = Kreyszig
| first = Erwin
| title = Differential geometry
| publisher = Dover Publications Inc.
| location = New York
| year = 1991
| pages = xiv+352
| isbn = 0-486-66721-9
| mr=1118149}}

==External links==
* {{MathWorld|urlname=GeodesicMapping|title=Geodesic mapping}}

[[Category:Differential geometry]]
[[Category:Geodesic (mathematics)]]

Geodesic map - Revision history

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