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609:) 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
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566:-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.--
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I thought it was very confusing to boldface "stripe," as if it's some standard terminology to be learned, only to say that the correct terminology is "plaque." So I un-boldfaced stripe. Also, these guys are p-dimensional, not n-p - dimensional, rrright? They're sets where x (which consists of n-p
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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,
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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
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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
274:. 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
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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.
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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
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Right, but then you added a link to it, which I thought was very strange. I did entertain the notion it might be a technical term though and that's why I wrote my comment in my edit description. I removed the link to avoid possible future confusion, since I figured you were using the word
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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.
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responsibility to stop them making some hypothetical mistake, which might distract some hypothetical reader some day. In fact that may remove some information they should have. The wiki isn't perfect; but the policy is that we assume editors are reasonable people first.
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I wasn't removing it because it was a red link. Let me emphasize: I removed it to avoid future confusion. I shudder to think of somebody clicking on it and getting sent to a page about a candy store. For example, try
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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.
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Should the line 'There is a global foliation theory, because topological constraints exist.' be 'There is no global foliation theory, because topological constraints exist.' ??
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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.
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My comment on the introduction paragraph. Obviously I used 'stripes' because it is a very quick way to get a visual image of a foliation, locally, for anyone.
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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.
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Foliations can be viewed as solutions to first order linear homogeneous PDEs -- look at the technique of the proof of the
Frobenius theorem.
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Therefore, I changed the lead, giving first the general mathematical definition, and then the more special relativistic definition.
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651:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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are both examples of foliations. If I understand this correctly, then foliations are just ways of decomposing an
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I'm getting a sense that this subject can be explained in a much less technical way with a few examples.
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I changed codimension by dimension, because I think that p is used in that sense in the definition.
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Is there someone among the principal contributors to this article willing to straighten this out?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I read somewhere that foliation theory was developed to understand the phase space structure of
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There is a mistake R^n must be a R^p when at the beginning you define what a foliation is.
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being the dimension of the ambient (embedding) manifold. I made this clear in the lead.
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I think that C class is closer to the current state of the article so I promoted it.
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Last edited at 13:42, 29 July 2009 (UTC). Substituted at 02:07, 5 May 2016 (UTC)
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foliation is understood as a stack of disjoint embedded hypersurfaces (dimension
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More on the definition: There seems to be some confusion about the domain of
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is the dimension of the leaves and also the dimension of the foliation, and
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responsibility to make sure the backlinks from it are all useful. It is not
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Sorry for replying here, not up with the interface quirks...
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I had a wild dream of having
Foliation be a redirect page to
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those in later parts. I hope that I am not misunderstood...
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and this page, which deals with mathematics, being called
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