Transformations and non-degenerate maps

摘要

We shall prove the equivalences of a non-degenerate circle-preserving map and a Moebius transformation in R~n, of a non-degenerate geodesic-preserving map and an isometry in H~n, of a non-degenerate line-preserving map and an affine transformation in R~n. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.