Talk:Liouville's theorem (conformal mappings)

Complex analysis??

Contrary to what some editors believe, this is not a theorem of complex analysis. Complex numbers do not even appear here. It's true that conformal mappings in two dimensions are complex-analytic (with respect to the unique complex structure compatible with a given conformal structure). But conformal mappings in higher dimensions have nothing to do with complex structures, and this theorem is specifically about dimension three and higher. Sławomir Biały (talk) 13:32, 15 February 2015 (UTC)

Today the link to Möbius transformation was removed. As our article indicates, Möbius mappings are usually confined to a plane. However, some authors note that the planar mappings are generated by multiplication, translation, and inversion and then proceed to generalize the term "Möbius mapping" to any mapping so generated. Such terminology confuses the subject and pushes the contributions of Herr Möbius beyond what he did while alive. Möbius made many contributions to mathematics and his legacy need not be magnified beyond his writings.Rgdboer (talk) 22:28, 16 February 2015 (UTC)

"Möbius mappings are usually confined to a plane". I'm not sure this is true. For example, volume 2 of Berger's Geometry considers Möbius transformations of the n-sphere. Moreover, many of the modern references on Liouville's theorem specifically refer to these as Möbius transformations. So it would arguably be more confusing not to use the standard terminology. I've not myself looked into what Möbius actually wrote, but there are lots of very high-quality secondary sources that use the term in this generalized sense. Sławomir Biały (talk) 02:24, 17 February 2015 (UTC)

Yes, I agree there is some of that usage of the term, but here in Wikipedia the term is restricted to the complex plane. Should we provide a section on Generalization of the term, then two directions correspond to the use of the Kelvin transform for Eⁿ and for ring theory the linear fractional transformations where the inversion corresponds to multiplicative inverse. The deletion of reference to Mobius today was to support your defense of this article from its inclusion in complex analysis. If you look at the article on Mobius transformations it will be clear why there has been an effort to put Liouville's theorem in that category. Perhaps this article is the place articulate the generalized conception, since, as you indicate, the references for this theorem use the term.Rgdboer (talk) 03:15, 17 February 2015 (UTC)

Liouville does refer to Kelvin's work, so "Kelvin transform" would probably be most correct. But no one calls it that, and I think it would be very confusing to refer to that in the article. I've looked through the literature, and the standard word for what we mean here carries the name Möbius. It goes back at least to Hartman's 1947 paper on Liouville's theorem [1], although other use of the term "Möbius transformation" to refer to conformal self-maps of the n-sphere appears earlier in works of Yano and Sasaki. Sławomir Biały (talk) 12:10, 17 February 2015 (UTC)

Yes, I see page 331 of Hartman (1947). With other references we have material for describing Liouville/Mobius mappings. The link to Möbius transformation should be removed because it misleads the reader of this article. The gap can be filled in with a special section explaining the generalized terminology in this article.Rgdboer (talk) 20:24, 17 February 2015 (UTC)

Ok, that seems like the best solution. Sławomir Biały (talk) 20:35, 17 February 2015 (UTC)

I've moved the digression as off-topic here. We have good sources that apply the term Möbius transformation (or Möbius group, Möbius geometry, etc) to refer to the n-dimensional case. These sources are not just specific to some niche use either, but in fact it is very widely used in this connection. It is much more appropriate to cover this wide use in the main article Möbius transformation, perhaps towards the end. I note also that there is an apparent content fork of that article also at conformal geometry#Möbius geometry. Sławomir Biały (talk) 12:27, 18 February 2015 (UTC)

Thank you for making the adjustments, especially the link to Conformal geometry, which somehow got off my Watchlist.Rgdboer (talk) 00:37, 19 February 2015 (UTC)