To-do list for Manifold/old: Resolve suggested article improvements presented in talk below:
Resolved
|
Manifold recently underwent a rather thorough rewriting, by
MarSch. The diff is
here.
Let me summarize how the article changed (but I would request you read the diffs for yourselves).
Can I say the new version is certainly bad? Probably no. Is it more rigurous and stresses more on heavy use of manifolds in math and physics? Yes. Is it more understandable for non-specialists? No. Is this article still a fine encyclopedic essay? Maybe, but worse than before. I would like to discuss if we maybe could move to the older version.
And I couple of words about MarSch. MarSch is a very smart guy, knowing lots of differential geometry, and probably many other things. MarSch is a very bold guy, having rewritten such a prominent article as mathematics (I did not see how it went), and MarSch desired to edit the main Wikipedia page, but lacked administrative powers to do that. MarSch has very good technical skills, and likes the technical parts to be more prominent in articles (there was a discussion right here about [[Laplace operator]). My own view is that MarSch needs to still absorb a bit more the existing culture on Wikipedia before making drastic changes. Oleg Alexandrov 19:16, 22 May 2005 (UTC)
moved my comment down -- MarSch 17:14, 4 Jun 2005 (UTC)
Lost in the debate was my primary objections to MarSch's edits. First was cutting the "intrinsic vs extrinsic part"; that still to be discussed. The second was the "cleanup" taken by MarSch as result of which charts were defined first and the treatment of topological manifolds and differential manifolds were rather combined. Now I reverted that to the way it was before. More work is needed, but this is a better starting point. Oleg Alexandrov 21:27, 26 May 2005 (UTC)
I would like to see Atlas (topology) expanded; the layman's intro is good, but a good mathematical definition would be a lot better. e.g. a discussion of the transition functions, and a discussion of the Jacobi identity-like thing (the Bianchi identity-like thing?) or whatever its called where you have three open sets intersecting, and the charts have to be three-way consistent. Maybe MarSch could apply himself there? linas 05:35, 2 Jun 2005 (UTC)
Should we allow (topological) manifolds to be empty? The current definition does allow this, but is contradicted by the following paragraph. The classification of 1-manifolds further down the page assumes they are nonempty, and there are other articles that also assume manifolds are nonempty. The only statement I have seen that relies on empty manifolds is the one about the boundary of an n-manifold being an an (n-1)-manifold. -- Zundark 16:25, 1 Jun 2005 (UTC)
Article states: (Readers should see the topology glossary for definitions of topological terms used in this article.) Can we cut this sentence out? linas 05:23, 2 Jun 2005 (UTC)
Article states: ...it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.
These sentences seem to imply that the "real line with two origins" is homeomorphic to the real line. It is not clear that it is, or that it isn't. Wording should be fixed. linas 05:23, 2 Jun 2005 (UTC)
Just an idea, but maybe the article should provide a few sentances on differential geometry and calculus on manifolds, maybe mentioning that a manifold has concepts of area and volume? linas 06:06, 2 Jun 2005 (UTC)
Wording needs to be fixed and clarified. For a Ck manifold M, you can form the Ck functions to the real or complex numbers. What does "..to the.." mean? Which Ck functions are these? Are these the charts? Are these some other functions defned on the manifold? Its known that the set of scalar-valued functions on a manifold form an algebra, is that what this paragraph is talking about? linas 06:06, 2 Jun 2005 (UTC)
Should this article mention (probably in the "see also" section) algebraic variety ? Or at least algebraic geometry? linas 06:06, 2 Jun 2005 (UTC)
I find this section in the article not very helpful. I mean, all the other sections talk about really important concepts about manifolds (tangent space, classification); this one I think is a minor thing, and I belive we could be better off without it. Comments? Oleg Alexandrov 16:52, 4 Jun 2005 (UTC)
I put this in here, because of the possibility to recover from the algebra of scalars the manifold itself. Unfortunately I don't know very much about this yet, specifically under what conditions this is possible. Also another def of the tangent bundle is as derivations on the algebra of scalars. In that way it is more basic than the tangent bundle, so I think that this is also "really important".
As an aside: It is the Wikipedia way to try to preserve all information. Even if you think something of minor importance, you should try to preserve it. Please try to be a bit more carefull in your suggestions about "things we can do without". -- MarSch 11:36, 5 Jun 2005 (UTC)
I would like to alert some editors to this template, which expresses the beliefs of a lot of wikipediansa about technical details:
{{
Technical|date=September 2010}}
That said, I would like to start trying to sort out our differences. Therefore I will start simple.
I have tried to integrate the chart-def with the homeomorphic neighborhood-def and also tried to explain the defs more. See below:
Topological manifold without boundary
The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn. Another way of saying this, using charts, is that a manifold without boundary is a topological space in which at every point there is an Rn-chart.
Topological manifold with boundary
More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is Euclidean half-space. Most points in Euclidean half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean half-space. Thus a topological manifold with is a topological space in which at each point there is an Rn-chart or an R0+×Rn-1-chart. The set of points at which there are only R0+×Rn-1-charts is called the boundary and its complement is called the interior.
I think the thing Oleg and linas didn't like about my previous changes was that I removed the section titled technical description. I've removed the sentence about what follows below, because that is not usefull at all. I think this section is very bad. It is not technical which it claims. It includes some history and some other random facts. Can we relocate all info and then delete this then empty section?
I've found two subsections which Oleg deleted by doing his partial revert. I've put them back. This does not mean that I want these as they are now, but we shouldn't be throwing the info out. The first is the section on Hausdorffness. Since Hausdorffness is no longer in the core def we might want to mention that most mathematicians do include Hausdorffness in the def. And second-countability also maybe? Not sure. The other section is random facts on (diff.) manifolds, ah I see there something about 2nd countability.
I've put in the expanded def. on topological manifolds. We should probably have something similar for diff. manifolds. Both a def with substitutions of subdefs and a short def. We should collect all random facts into either the section about topo or about diff manifolds. It will probably turn out that diff manifold could be split off. Maybe we should also split topo manifold off. We could then expand on some more general manifolds like the Banach and Frechet manifolds. Maybe a section for Lie groups. And of course a section about complex manifolds. Also needed is the canonical picture for clarifying what a transition map is. Tell me what you think -- MarSch 14:45, 11 Jun 2005 (UTC)
Well, that depends on what you call a non-detail. We need to be more specific. Here is a proposal of how I'd like to see the page:
Comments? -- Jitse Niesen 21:07, 22 Jun 2005 (UTC)
I like very much wat Jitse suggests. Pictures needed! I can make pictures like the one in level set method, but I don't know if that's the best style. Anyway, what kind of pictures to put? Oleg Alexandrov 02:51, 23 Jun 2005 (UTC)
Charts are introduced in Technical description, and we can discuss them more fully in chart (topology) or perhaps topological manifold. I agree that charts are essential in manifold theory, but I do not want much manifold theory in the article. I am undecided as of yet whether we want any definitions of topological and differentiable manifolds in the article, or whether this would complicate things too much. I will hopefully find time to flesh it out a bit more later, but for the first paragraph of differentiable manifolds, I am thinking along the lines of the current first paragraph, which goes "It is easy to define the notion of a topological manifold, but it is very hard to work with this object. The smooth manifold defined below works better for most applications, in particular it makes possible to apply "calculus" on the manifold." -- Jitse Niesen 12:26, 23 Jun 2005 (UTC)
I had a wee go at it last night, but it is not easy. You can see my efforts in User:Jitse Niesen/sandbox. It's very much work in progress, but you're free to edit. Actually, I now think that charts and transition maps should be described, as they are not difficult concepts. We might be able to steal some pictures from the Japanese article at ja:多様体. -- Jitse Niesen ( talk) 13:00, 25 Jun 2005 (UTC)
I just want to warn you not to hold your breath. I wanted to work on it last night, but didn't manage, and I'll be away for a week. Ideally, you have everything finished when I return, otherwise, I may have another go at it myself. -- Jitse Niesen ( talk) 23:59, 25 Jun 2005 (UTC)
I believe the rewrite alert box to be way too strident to show up on the article page. Let us keep some appearances here; the article page should look like a polished page and not some scratch pad. I put the alert box on this talk page. I highly doubt that the lack of feedback about the rewrite is due to the fact that I removed the ugly alert box a couple of days ago and again today. Oleg Alexandrov 29 June 2005 16:11 (UTC)
Well regardless of whether improving articles is more important than appearances, it is a long-established and well-accepted practice that editorial comments belong on the talk page not on the article page. Paul August ☎ June 30, 2005 20:15 (UTC)
For that matter, a template inserted in an article should not have a fat red frame, rather be as unnoticeable as possible while still doing the job. Oleg Alexandrov 7 July 2005 16:09 (UTC)
Any of you maths genii feel like writing some kind of article on Frölicher spaces? I would, but as soon as I see math where numbers start getting replaced by letters, I break out in cold sweat. I only ask cause it's on the list of Articles requested for more than two years. Ta. Proto t c 4 July 2005 12:44 (UTC)
This link is a start. Check the bottom of page 16 for the definition. It seems to be a part of synthetic differential geometry. See this for the associated philosophy. - Gauge 21:05, 10 July 2005 (UTC)
Congrats to the cooks! After a quick glance, the article looks great! No doubt, once I read it carefully, I'll find plenty o' things awry, but from the distance, it looks nice. linas 7 July 2005 00:29 (UTC)
Please insert a layperson definition of a manifold. if you can understand the technical description below, you probably already know what one is... --anon user (moved here by Oleg Alexandrov 23:05, 28 July 2005 (UTC))
It looks to me that Deryck Chan copied over manifold/rewrite to manifold, and without any discussion or edit summary. Is that the right thing to do? Oleg Alexandrov 00:24, 4 August 2005 (UTC)
Despite the fact that the talk page prominently says a rewrite is in progress, some folks have continued to edit this page. Either they are wasting their time, or someone is going to have to try to integrate their edits into the rewrite — and it ain't gonna be me! Thus the warning language spells it out unambiguously, so no one can complain when the edits are lost. Parallel efforts make no sense. KSmrq 22:05, 2005 August 6 (UTC)
Therefore, as I say, we should NOT have a separate rewrite page. Deryc k C. 09:34, 7 August 2005 (UTC)
Sorry. I did not mean it. Oleg Alexandrov 15:36, 8 August 2005 (UTC)
I applaude the effort, but if you don't want the page edited, why not request to have it locked? And why not just do a straight merge? I can do this at any point if so requested. - Ta bu shi da yu 04:18, 9 August 2005 (UTC)
The rewrite is at a separate page, because it is felt by some that it is important that the entry is in a polished finished state. The rewrite frequently (used to have?) unpolished and unfinished stuff and even scaffolding. The separaration allows us to work more flexibly and allowed us to copy in content from here while it was still lacking. I don't really care about it either way as long as the article keeps improving. -- MarSch 13:11, 14 August 2005 (UTC)
I have marked this page for deletion - subpages do not belong, and are in fact disabled, in the main article space. I landed here by clicking random article, yet this is not an article, but an archived version of one. If it is meant to be a freestanding article, that represents a content fork from Manifold, which should not exist. If it is here merely as a resource for people working on Manifold, it should exist only in user or talk namespace, not in the main article namespace. I am posting here to alert anyone who may care about this content to move it to a different namespace before it is deleted. — Swpb talk contribs 06:42, 17 January 2007 (UTC)
To-do list for Manifold/old: Resolve suggested article improvements presented in talk below:
Resolved
|
Manifold recently underwent a rather thorough rewriting, by
MarSch. The diff is
here.
Let me summarize how the article changed (but I would request you read the diffs for yourselves).
Can I say the new version is certainly bad? Probably no. Is it more rigurous and stresses more on heavy use of manifolds in math and physics? Yes. Is it more understandable for non-specialists? No. Is this article still a fine encyclopedic essay? Maybe, but worse than before. I would like to discuss if we maybe could move to the older version.
And I couple of words about MarSch. MarSch is a very smart guy, knowing lots of differential geometry, and probably many other things. MarSch is a very bold guy, having rewritten such a prominent article as mathematics (I did not see how it went), and MarSch desired to edit the main Wikipedia page, but lacked administrative powers to do that. MarSch has very good technical skills, and likes the technical parts to be more prominent in articles (there was a discussion right here about [[Laplace operator]). My own view is that MarSch needs to still absorb a bit more the existing culture on Wikipedia before making drastic changes. Oleg Alexandrov 19:16, 22 May 2005 (UTC)
moved my comment down -- MarSch 17:14, 4 Jun 2005 (UTC)
Lost in the debate was my primary objections to MarSch's edits. First was cutting the "intrinsic vs extrinsic part"; that still to be discussed. The second was the "cleanup" taken by MarSch as result of which charts were defined first and the treatment of topological manifolds and differential manifolds were rather combined. Now I reverted that to the way it was before. More work is needed, but this is a better starting point. Oleg Alexandrov 21:27, 26 May 2005 (UTC)
I would like to see Atlas (topology) expanded; the layman's intro is good, but a good mathematical definition would be a lot better. e.g. a discussion of the transition functions, and a discussion of the Jacobi identity-like thing (the Bianchi identity-like thing?) or whatever its called where you have three open sets intersecting, and the charts have to be three-way consistent. Maybe MarSch could apply himself there? linas 05:35, 2 Jun 2005 (UTC)
Should we allow (topological) manifolds to be empty? The current definition does allow this, but is contradicted by the following paragraph. The classification of 1-manifolds further down the page assumes they are nonempty, and there are other articles that also assume manifolds are nonempty. The only statement I have seen that relies on empty manifolds is the one about the boundary of an n-manifold being an an (n-1)-manifold. -- Zundark 16:25, 1 Jun 2005 (UTC)
Article states: (Readers should see the topology glossary for definitions of topological terms used in this article.) Can we cut this sentence out? linas 05:23, 2 Jun 2005 (UTC)
Article states: ...it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.
These sentences seem to imply that the "real line with two origins" is homeomorphic to the real line. It is not clear that it is, or that it isn't. Wording should be fixed. linas 05:23, 2 Jun 2005 (UTC)
Just an idea, but maybe the article should provide a few sentances on differential geometry and calculus on manifolds, maybe mentioning that a manifold has concepts of area and volume? linas 06:06, 2 Jun 2005 (UTC)
Wording needs to be fixed and clarified. For a Ck manifold M, you can form the Ck functions to the real or complex numbers. What does "..to the.." mean? Which Ck functions are these? Are these the charts? Are these some other functions defned on the manifold? Its known that the set of scalar-valued functions on a manifold form an algebra, is that what this paragraph is talking about? linas 06:06, 2 Jun 2005 (UTC)
Should this article mention (probably in the "see also" section) algebraic variety ? Or at least algebraic geometry? linas 06:06, 2 Jun 2005 (UTC)
I find this section in the article not very helpful. I mean, all the other sections talk about really important concepts about manifolds (tangent space, classification); this one I think is a minor thing, and I belive we could be better off without it. Comments? Oleg Alexandrov 16:52, 4 Jun 2005 (UTC)
I put this in here, because of the possibility to recover from the algebra of scalars the manifold itself. Unfortunately I don't know very much about this yet, specifically under what conditions this is possible. Also another def of the tangent bundle is as derivations on the algebra of scalars. In that way it is more basic than the tangent bundle, so I think that this is also "really important".
As an aside: It is the Wikipedia way to try to preserve all information. Even if you think something of minor importance, you should try to preserve it. Please try to be a bit more carefull in your suggestions about "things we can do without". -- MarSch 11:36, 5 Jun 2005 (UTC)
I would like to alert some editors to this template, which expresses the beliefs of a lot of wikipediansa about technical details:
{{
Technical|date=September 2010}}
That said, I would like to start trying to sort out our differences. Therefore I will start simple.
I have tried to integrate the chart-def with the homeomorphic neighborhood-def and also tried to explain the defs more. See below:
Topological manifold without boundary
The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn. Another way of saying this, using charts, is that a manifold without boundary is a topological space in which at every point there is an Rn-chart.
Topological manifold with boundary
More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is Euclidean half-space. Most points in Euclidean half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean half-space. Thus a topological manifold with is a topological space in which at each point there is an Rn-chart or an R0+×Rn-1-chart. The set of points at which there are only R0+×Rn-1-charts is called the boundary and its complement is called the interior.
I think the thing Oleg and linas didn't like about my previous changes was that I removed the section titled technical description. I've removed the sentence about what follows below, because that is not usefull at all. I think this section is very bad. It is not technical which it claims. It includes some history and some other random facts. Can we relocate all info and then delete this then empty section?
I've found two subsections which Oleg deleted by doing his partial revert. I've put them back. This does not mean that I want these as they are now, but we shouldn't be throwing the info out. The first is the section on Hausdorffness. Since Hausdorffness is no longer in the core def we might want to mention that most mathematicians do include Hausdorffness in the def. And second-countability also maybe? Not sure. The other section is random facts on (diff.) manifolds, ah I see there something about 2nd countability.
I've put in the expanded def. on topological manifolds. We should probably have something similar for diff. manifolds. Both a def with substitutions of subdefs and a short def. We should collect all random facts into either the section about topo or about diff manifolds. It will probably turn out that diff manifold could be split off. Maybe we should also split topo manifold off. We could then expand on some more general manifolds like the Banach and Frechet manifolds. Maybe a section for Lie groups. And of course a section about complex manifolds. Also needed is the canonical picture for clarifying what a transition map is. Tell me what you think -- MarSch 14:45, 11 Jun 2005 (UTC)
Well, that depends on what you call a non-detail. We need to be more specific. Here is a proposal of how I'd like to see the page:
Comments? -- Jitse Niesen 21:07, 22 Jun 2005 (UTC)
I like very much wat Jitse suggests. Pictures needed! I can make pictures like the one in level set method, but I don't know if that's the best style. Anyway, what kind of pictures to put? Oleg Alexandrov 02:51, 23 Jun 2005 (UTC)
Charts are introduced in Technical description, and we can discuss them more fully in chart (topology) or perhaps topological manifold. I agree that charts are essential in manifold theory, but I do not want much manifold theory in the article. I am undecided as of yet whether we want any definitions of topological and differentiable manifolds in the article, or whether this would complicate things too much. I will hopefully find time to flesh it out a bit more later, but for the first paragraph of differentiable manifolds, I am thinking along the lines of the current first paragraph, which goes "It is easy to define the notion of a topological manifold, but it is very hard to work with this object. The smooth manifold defined below works better for most applications, in particular it makes possible to apply "calculus" on the manifold." -- Jitse Niesen 12:26, 23 Jun 2005 (UTC)
I had a wee go at it last night, but it is not easy. You can see my efforts in User:Jitse Niesen/sandbox. It's very much work in progress, but you're free to edit. Actually, I now think that charts and transition maps should be described, as they are not difficult concepts. We might be able to steal some pictures from the Japanese article at ja:多様体. -- Jitse Niesen ( talk) 13:00, 25 Jun 2005 (UTC)
I just want to warn you not to hold your breath. I wanted to work on it last night, but didn't manage, and I'll be away for a week. Ideally, you have everything finished when I return, otherwise, I may have another go at it myself. -- Jitse Niesen ( talk) 23:59, 25 Jun 2005 (UTC)
I believe the rewrite alert box to be way too strident to show up on the article page. Let us keep some appearances here; the article page should look like a polished page and not some scratch pad. I put the alert box on this talk page. I highly doubt that the lack of feedback about the rewrite is due to the fact that I removed the ugly alert box a couple of days ago and again today. Oleg Alexandrov 29 June 2005 16:11 (UTC)
Well regardless of whether improving articles is more important than appearances, it is a long-established and well-accepted practice that editorial comments belong on the talk page not on the article page. Paul August ☎ June 30, 2005 20:15 (UTC)
For that matter, a template inserted in an article should not have a fat red frame, rather be as unnoticeable as possible while still doing the job. Oleg Alexandrov 7 July 2005 16:09 (UTC)
Any of you maths genii feel like writing some kind of article on Frölicher spaces? I would, but as soon as I see math where numbers start getting replaced by letters, I break out in cold sweat. I only ask cause it's on the list of Articles requested for more than two years. Ta. Proto t c 4 July 2005 12:44 (UTC)
This link is a start. Check the bottom of page 16 for the definition. It seems to be a part of synthetic differential geometry. See this for the associated philosophy. - Gauge 21:05, 10 July 2005 (UTC)
Congrats to the cooks! After a quick glance, the article looks great! No doubt, once I read it carefully, I'll find plenty o' things awry, but from the distance, it looks nice. linas 7 July 2005 00:29 (UTC)
Please insert a layperson definition of a manifold. if you can understand the technical description below, you probably already know what one is... --anon user (moved here by Oleg Alexandrov 23:05, 28 July 2005 (UTC))
It looks to me that Deryck Chan copied over manifold/rewrite to manifold, and without any discussion or edit summary. Is that the right thing to do? Oleg Alexandrov 00:24, 4 August 2005 (UTC)
Despite the fact that the talk page prominently says a rewrite is in progress, some folks have continued to edit this page. Either they are wasting their time, or someone is going to have to try to integrate their edits into the rewrite — and it ain't gonna be me! Thus the warning language spells it out unambiguously, so no one can complain when the edits are lost. Parallel efforts make no sense. KSmrq 22:05, 2005 August 6 (UTC)
Therefore, as I say, we should NOT have a separate rewrite page. Deryc k C. 09:34, 7 August 2005 (UTC)
Sorry. I did not mean it. Oleg Alexandrov 15:36, 8 August 2005 (UTC)
I applaude the effort, but if you don't want the page edited, why not request to have it locked? And why not just do a straight merge? I can do this at any point if so requested. - Ta bu shi da yu 04:18, 9 August 2005 (UTC)
The rewrite is at a separate page, because it is felt by some that it is important that the entry is in a polished finished state. The rewrite frequently (used to have?) unpolished and unfinished stuff and even scaffolding. The separaration allows us to work more flexibly and allowed us to copy in content from here while it was still lacking. I don't really care about it either way as long as the article keeps improving. -- MarSch 13:11, 14 August 2005 (UTC)
I have marked this page for deletion - subpages do not belong, and are in fact disabled, in the main article space. I landed here by clicking random article, yet this is not an article, but an archived version of one. If it is meant to be a freestanding article, that represents a content fork from Manifold, which should not exist. If it is here merely as a resource for people working on Manifold, it should exist only in user or talk namespace, not in the main article namespace. I am posting here to alert anyone who may care about this content to move it to a different namespace before it is deleted. — Swpb talk contribs 06:42, 17 January 2007 (UTC)