3 ms·
Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations.
by jacobolus 1y ago
Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations. The 2 dimensional case is special.
See: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings) https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...
- srean 1y agoYes of course, but I am curious about any interesting structures that functions from quaternion to quaternion may possess. I used conformal mapping as an example of an interesting structure. I could have used Cauchy Riemann as another example.
- jacobolus 1y agoYou're probably looking for something like Sudbery 1977, https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-analysis-memo.pdf https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana... (published 1979, doi: 10.1017/S0305004100055638)
- srean 1y agoFantastic. Thanks for the reference.