Talk:Conformal geometry

Expert attention

I added the expert-subject template because I see some problems with this article that I don't feel competent enough to remedy.

The article is divided into two seemingly disjoint parts on "conformally flat geometry" and "conformally curved geometry" (including an offhand remark that the practioners of the latter call it "conformal geometry"). The following Google hit counts suggest that this terminology is not in wide usage:

"conformally curved geometry": 9

"conformally flat geometry": 222

"conformally curved": 309

"conformally flat": 43500

It seems to me that "conformally curved geometry" should be called something like "general conformal geometry", since of course it also allows conformally flat spaces as a special case.

The part on "conformally flat geometry" then makes a further distinction between "two dimensions" and "higher dimensions", which suffers from the same problem in that at least some of the things under "higher dimensions" seem to apply in the general case.

The concept of a "conformal class" or "conformal structure" (an equivalence class of Riemannian metrics under conformal equivalence) is general and should feature prominently in the article. (Both of these terms redirect to this article). Currently this is introduced separately and differently in the two parts -- as "conformal class" in a subsection of the "higher dimensions" section of the part on "conformally flat geometry", and as "conformal structure" in the part on "conformally curved geometry".

The term "conformal manifold" (a manifold equipped with a conformal structure, i.e. with an equivalence class of Riemannian metrics) should also be introduced. Currently the part on "conformally curved geometry" takes a round-about approach by first saying that "conformally curved geometry" studies a (pseudo-)Riemannian manifold with metric g and then somewhat non-rigorously saying that this metric is "only defined up to scale"; later on it uses the term "conformal manifold" without having defined it. It seems clearer to start out by defining conformal classes/structures and then saying that conformal geometry studies conformal manifolds, i.e. manifolds equipped with a conformal structure. I set up a redirect to this article for "conformal manifold", which doesn't have its own entry (perhaps it should?).

The term "conformal space" redirects here; I'm not sure whether this is a synonym for "conformal manifold".

I'd be happy to help with all this, but I'm not an expert and am not sure exactly which parts really do belong in specific sections on "conformally flat geometry" and "higher dimensions".

Joriki (talk) 05:58, 10 April 2009 (UTC)

The word "conformally flat" is in wide usage. I'm not sure how to address the terminological conundrum you raise, though. Sometimes what the article calls "conformally flat geometry" is known as Möbius geometry. In that case, "conformally curved geometry" is typically known as just "conformal geometry", but this risks confusion and there seems to be no standard way to disambiguate between the two notions. I'll do what I can. Sławomir Biały (talk) 15:05, 20 June 2009 (UTC)

I have standardized on Mobius geometry for the flat case. You make an excellent point that the article could place more emphasis on conformal structures earlier on. I don't really know how to do this and leave the article intact. But I have dealt with what I think are the more urgent issues in your post, and removed the {{expert}} tag. Sławomir Biały (talk) 15:29, 20 June 2009 (UTC)

Conformally flat and locally conformally flat

There is a mistake in this article concerning the definition of conformally flat and locally conformally flat. These are not the same. LCF means LOCALLY conformally equivalent to a flat Riemannian manifold. CF means GLOBALLY conformally equivalent to a flat manifold. The round sphere in dimensions 2 and up are examples of LCF manifolds which are not CF. In fact any constant curvature manifold is LCF, but is certainly not CF.Geminatea (talk) 22:51, 17 September 2010 (UTC)

In the context of conformal geometry as it is addressed in this article (primarily via transformation groups), "conformally flat" usually means that the manifold is conformally isometric to a subset of the conformal sphere (see, for instance, Akivis and Goldberg "Conformal geometry and its generalizations). This usage differs somewhat from that in geometric analysis: geometric analysis always rigidly insist that "conformally flat" means (effectively) covered by a subset of Euclidean space. To add still more confusion to the mix, for most physicists and many differential geometers, "conformally flat" means what the geometric analysis insist should be called "locally conformally flat". That said, I have edited in a manner to preserve the spirit of your "correction", although I don't think it improves the clarity of the article. Sławomir Biały (talk) 14:24, 11 January 2011 (UTC)

Definition of preserving the quadric

Under Conformal geometry#The projective model, does preserving the quadric mean it's invariant under the projective transformation? ᛭ LokiClock (talk) 01:03, 24 February 2013 (UTC)

Yes, specifically in the sense of Group action#Invariant subsets. Sławomir Biały (talk) 12:22, 25 February 2013 (UTC)

Thanks! I'm changing the language, because "preserving" makes it sound like it's a structure the space is equipped with. With the quadratic sitting right next to it, it's aching for a misunderstanding. ᛭ LokiClock (talk) 02:04, 26 February 2013 (UTC)

Mobius Geometry / Conformal Geometry

I am a bit confused about the content of this article, not that it is wrong but that (to me) Mobius geometry is more the 1D-down subgeometry of Lie Sphere Geometry (which has its own article). I don't really see inversive geometry as a model of Mobius geometry for instance although projective certainly is. Shouldn't it be two articles, one for conformal geometry and the other for Mobius geometry? Or is this one of those wrangles where different branches have claims on the same terminology? Selfstudier (talk) 20:03, 31 October 2017 (UTC)

Or else accepting the division into flat and curved, with flat being Mobius geometry, then I guess the problem I am having is the inclusion of the low dimensional, specifically the 2d stuff, as being Mobius geometry. Let me see if I can find some references and then go from there. Selfstudier (talk) 12:43, 1 November 2017 (UTC)

I think I am starting to get it now, apart from the flat/curved split, there is another split as between 2D and 3D + with conformal maps for the latter being governed by the Liouville's_ theorem (conformal mappings) (even though the transformations are in both cases called Möbius transformation#Higher dimensions). I don't think we have really got this clearly in the article.Selfstudier (talk) 13:13, 1 November 2017 (UTC)

OK, I am not going to alter anything until I have the refs for it.Selfstudier (talk) 13:17, 1 November 2017 (UTC)