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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 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. * 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「geodesic map」の詳細全文を読む スポンサード リンク
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