A characterization of Möbius transformations

Abstract

Let n ≥ 2 n\ge 2 be an integer and let D \mathcal {D} be a domain of R n \mathbb {R}^n . Let f : D → R n f:\mathcal {D}\to \mathbb {R}^n be an injective mapping which takes hyperspheres whose interior is contained in D \mathcal {D} to hyperspheres in R n \mathbb {R}^n . Then f f is the restriction of a Möbius transformation.