On the circle preserving property of Möbius transformations

Abstract

This paper is mainly concerned with the study of circle-preserving property of Möbius transformations acting on. The circle-preserving property is the most known invariant characteristic property of Möbius transformations. Obviously, a Möbius transformation acting on is circle-preserving. Recently, for the converse statement, some interesting and nice results have been obtained. Here, we investigate these studies. We consider the relationships between Möbius transformations and sphere-preserving maps in since the studies about the circle-preserving property of maps in are related to the study of sphere-preserving maps. For the case n∈=∈2, we also consider the problem whether or not the circle-preserving property is an invariant characteristic property of Möbius transformations for the circles corresponding to any norm function on.