3

Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has $n+3$ distinct points any $n+2$ of which are in general position (in the sense of not being contained in an $(n-1)$-sphere), then $f$ is a Möbius transformation.

Here, we make no any other assumption on $f$, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.

Jukka Kohonen
  • 3,964
  • 2
  • 19
  • 49
woodbass
  • 425
  • 2
    This theorem is well-known in certain circles. – Ian Agol Dec 30 '12 at 18:28
  • Can you provide the source of the well-known theorem? – woodbass Dec 30 '12 at 18:35
  • I imagine this is a follow-up to http://mathoverflow.net/questions/117436/a-problem-about-spherical-transformation-circle-mapping – alvarezpaiva Dec 30 '12 at 18:43
  • Oh, you're not assuming continuous - usually the term "map" refers to a continuous function. I don't know a reference, I think I worked this out when I was a graduate student by showing that it takes spheres to spheres, and induction on dimension. – Ian Agol Dec 30 '12 at 18:45
  • @Agol: Do you mean that you proved the conjecture under the assumption that f is continuous? If no assumption on continuity, do you have any idea or know any results in literature? – woodbass Dec 30 '12 at 18:58
  • 1
    Not an answer, but see Caratheodory, "The most general transformations of plane regions which transform circles into circles". He doesn't require continuity, but does require bijectivity in some region. Haven't read it, though. I was redirected to there by McCallum, "GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY". Also see Höfer, "A CHARACTERIZATION OF MOBIUS TRANSFORMATIONS", which deals with arbitrary dimension, but still requires injectivity in a region and that the map sends hyperspheres to hyperspheres. – Marcos Cossarini Dec 31 '12 at 05:27

0 Answers0