geodesic map

Every totally geodesic map is harmonic (in this case, dφh itself vanishes, not just its trace).

There is no geodesic map from the Euclidean space E onto the unit sphere S, since they are not homeomorphic, let alone diffeomorphic.

In mathematics-specifically, in differential geometry-a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics".

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.

If (M, g) and (N, h) are both the n-dimensional Euclidean space E with its usual flat metric, then any Euclidean isometry is a geodesic map of E onto itself.

In mathematics - specifically, in Riemannian geometry - Beltrami's theorem is a result named after the Italian mathematician Eugenio Beltrami which states that geodesic maps preserve the property of having constant curvature.

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 R, 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.