• φ is a diffeomorphism
  Diffeomorphism
  In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
  
   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
  Inverse function
  In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
  
   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
  Dimension
  In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
  
  al Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Eⁿ with its usual flat metric, then any Euclidean isometry
  Isometry
  In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
  
   is a geodesic map of Eⁿ onto itself.

• Similarly, if (M, g) and (N, h) are both the n-dimensional unit sphere
  Hypersphere
  In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
  
   Sⁿ with its usual round metric, then any isometry of the sphere is a geodesic map of Sⁿ onto itself.

• If (M, g) is the unit sphere Sⁿ with its usual round metric and (N, h) is the sphere of radius
  Radius
  In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
  
   2 with its usual round metric, both thought of as subsets of the ambient coordinate space Rⁿ⁺¹, then the "expansion" map φ : Rⁿ⁺¹ → Rⁿ⁺¹ given by φ(x) = 2x induces a geodesic map of M onto N.

• There is no geodesic map from the Euclidean space Eⁿ onto the unit sphere Sⁿ, since they are not homeomorphic
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  , let alone diffeomorphic.

• The gnomonic projection
  Gnomonic projection
  A gnomonic map projection displays all great circles as straight lines. Thus the shortest route between two locations in reality corresponds to that on the map. This is achieved by projecting, with respect to the center of the Earth , the Earth's surface onto a tangent plane. The least distortion...
  
   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
  Hyperbolic geometry
  In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
  
   metric as in the Poincaré disc model of hyperbolic geometry. Then, although the two structures are diffeomorphic via the identity map
  Identity function
  In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
  
   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.