Talk:Foliation

message: I had a wild dream of having Foliation be a redirect page to foliation (disambiguation) and this page, which deals with mathematics, being called foliation (mathematical), and the gelogical us being foliation (geology). But with over 40 pages linking here on the math side and only 4 on the geology side, it seems silly.
However, it is my request of fellow wikiers that the link to the diambiguation page be maintained at the top, as people interested in the geological usage can then go through to there and find foliation (geology. Its either that or someone will have to go the disambiguation route, which seems like a lot of work. 
Cheers,Rolinator 03:35, 30 December 2005 (UTC)
thread_title: foliation (geology)

message: Does anyone know about the history of foliation theory?  the book by Camacho and Neto says that it was invented as an attempt to solve Poincare's conjecture, but a theorem of Novikov shattered this hopes (every foliation of a closed 3-manifold with trivial, or finite?, fundamental group has a Reeb component).  This book, or any other I know of, does not provide a reason why study/ do research on this subject.  So, perhaps this entry would benefit if a reason to study foliations is provided.
:I read somewhere that foliation theory was developed to understand the phase space structure of dynamical systems. Perhaps I can dig up some material on the subject. MathMartin 18:39, 26 September 2006 (UTC)
replies: text: Foliations can be viewed as solutions to first order linear homogeneous PDEs -- look at the technique of the proof of the Frobenius theorem.  Rybu (talk) 16:51, 17 March 2020 (UTC)
thread_title:  History of the subject 

message: I changed codimension by dimension, because I think that p is used in that sense in the definition.
It is confusing what p and n-p are for. From the definition, the dimension of leaves (or strips/plaques), where x=constant, should be p-dimensional, not n-p dimensional as stated. But this is inconsistent with what in Examples and Foliation... One has to either switch p and n-p  in Definition, or those in later parts. I hope that I am not misunderstood...
:Nobody seems to react to this criticism and yet it points to a true problem in the text, at least in the version as of today. In the definition section x stands for the first n-p coordinates and leaves are identified with x=constant so leaves are p dimensional. In the section Foliations and integrability, the dimension of the integral curves of a vector field is first identified with the dimension of the leaves of a 1 dimensional foliation. However, in the next alinea, integral manifolds of an (n-p) dimensional distribution appear to be identified with the leaves of a foliation. 
:In the Examples section, the "book" example, although correct in itself, is not presented in the same logic as the definition, which adds to the confusion. 
:Is there someone among the principal contributors to this article willing to straighten this out? Bas Michielsen (talk) 22:59, 11 August 2008 (UTC)
replies: text: p is the dimension of the leaves and also the dimension of the foliation, and q = n - p is the codimension of the foliation, n being the dimension of the ambient (embedding) manifold. I made this clear in the lead. Lantonov (talk) 10:28, 27 August 2018 (UTC)
thread_title:  modification of the definition 

message: I'm getting a sense that this subject can be explained in a much less technical way with a few examples. Wolfram Mathematica's help documentation suggests that grid lines and contour lines are both examples of foliations. If I understand this correctly, then foliations are just ways of decomposing an n-dimensional manifold into a collection of (n-1)-dimensional manifolds. This could not be less obvious from reading the page. Such a simplified explanation should go either in the introduction or in a section before the more theoretic explanation presently included. I have not made such a change pending confirmation by someone who can confirm my interpretation.--Ipatrol (talk) 17:41, 16 June 2015 (UTC)
replies: text: lpatrol, maybe you could add a section and call it "A simple example: mesh functions". In it you or someone else could explain why not every foliation is this simple. Crasshopper (talk) 19:12, 18 July 2015 (UTC)
thread_title:  Excessively technical 

message: From what I understand in this article, by "foliation" it means covering the manifold with immersed submanifolds. Although this may be correct as a very general and abstract definition, it is not how foliation is understood in the disciplines that use it (mainly relativity). In such applied sense, foliation is understood as a stack of disjoint embedded hypersurfaces (dimension n - 1) with all the properties of embedding, such as homeomorphism, push-forward and pull-back functions, etc. Also, foliation involves some additional structures that connect the leaves, such as a common normal or a requirement for the hypersurfaces to be Cauchy surfaces. Such a definition would be more useful for the people which actually use the foliation concept, and they are primarily numerical relativists doing 3+1 decomposition of spacetime. So I confirm the interpretation of Ipatrol above. 
Therefore, I changed the lead, giving first the general mathematical definition, and then the more special relativistic definition.
Also, the difference between stripes, plaques, and leaves is not clear in the article. I intend to add more precise definitions of rectangular neighbourhood, foliated atlas, etc. and then the link to overlapping plaques of a foliated atlas to form the leaves of a foliation. 
Lantonov (talk) 07:51, 27 August 2018 (UTC)
thread_title:  Comments about the definition 

message: Substituted at 02:07, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I think that C class is closer to the current state of the article so I promoted it. Lantonov (talk) 10:48, 26 September 2018 (UTC)
thread_title: Start to C class