Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 Moebius 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.

share|improve this question
1  
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 mathoverflow.net/questions/117436/… –  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
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.