This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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:
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)
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)
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)
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)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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:
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)
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)
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)
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)