Talk:Manifold/Archive 5

This is an archive of past discussions about Manifold. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

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Archive 3

Strange visitor from another planet

fights a neverending battle for truth, justice, and the Wikipedia way. The point being that the battle is indeed neverending.

Over in Mathematics Oleg Alexandrov simply reverts anything that is not clearly a major improvement. But for that to work, the article has to be in very good shape to begin with, which I don't think this article is, now.

Remembering the history of the article, a lot of good work was reverted by a single Wikipedian who found a single source that allowed manifolds to have more than one dimension. That is now reflected in a sentence somewhere in the article, and we are trying to struggle back to the light. Rick Norwood 21:04, 26 April 2006 (UTC)

What the.... are you suggesting that you'd like this article to only (or even mostly) treat 1-dimensional manifolds? 1-dimensional manifolds are pretty boring, lots of stuff that manifold theory is useful for is completely trivial in 1-dimension. I'd like to see you find even a single source that doesn't allow manifolds with more than one dimension. -lethe ^{talk +} 21:10, 26 April 2006 (UTC)

There has been a misunderstanding here. Rick meant that the source allowed a manifold to have charts of different dimensions instead of requiring all charts to be of the same dimension. Loom91 06:40, 27 April 2006 (UTC)

Oh, I see. Thank you for clearing that up. Now that I know what Rick's complain really was, I'll say that I don't see any problem allowing for the fact that we may or may not require the dimension to be constant. It doesn't really get in the way or anything does it? -lethe ^{talk +} 06:58, 27 April 2006 (UTC)

Yet another attempt at an introduction which is both accurate and non-technical.

I'm going to begin this time by saying what I think we all want to say, and then trying to say it in non-technical languages.

First: a manifold is, in the sense we are talking about, a mathematical object. Second: It is a space (whatever that means) Third: It is Hausdorff and Second Countable (I think we can leave that for later). Finally: Every point has a neighborhood homeomorphic to (looks like) an n-ball.

In other words, I think we all agree on the correct mathematical definition of an n-manifold and that it has nothing to do with sticking pieces together or with close-ups.

Here is my proposed introduction, incorporating the changes suggested by Lethe and Septentrionalis, and now incorporating changes suggested by Loom91:

A manifold is a mathematical space in which every point has a neighborhood which looks like Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. Lines are one-dimensional, planes two-dimensional.

In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and a pair of circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.

Comments? Rick Norwood 22:44, 26 April 2006 (UTC)

You say that we all agree that it has nothing to do with either sticking pieces together or with close-ups. I don't agree, I think that's the essence of the definition. On some level, you must disagree as well, because you use the word "locally" in your description, which is nothing but a technical word for "at small scales" or "close-up". Speaking of which, I think perhaps the word "locally" might be off-putting and could be replaced by "at small scales" or something. I think we can all agree that Hausdorff and countability don't belong in the intro, yes. Finally, homeomorphic to a sphere? Probably you mean ball? I think it's a bit odd to define a manifold as locally homeomorphic to a sphere, since a sphere is topologically nontrivial. A space locally homeomorphic to a sphere would look pretty weird. Maybe the Hawaiian rings would satisfy that definition though? -lethe ^{talk +} 21:54, 26 April 2006 (UTC)

I don't think the minimal set of charts, which is what "sticking things together" usually implies, is the essence of a manifold; rather the maximal set, which I think Rob is thinking of. How about an approach with:

every point has a neighborhood which looks like a Euclidean n-space....

Septentrionalis 22:19, 26 April 2006 (UTC)

(Edit conflict) As I've admonished repeatedly, novices understand the word "space" in a very different sense than the mathematical convention. In fact, manifolds themselves are a natural route to understanding the way mathematicians use the term. The concept of "dimension" is far too advanced and sophisticated to lead with. We can have infinite dimension, and also different dimensions depending on how we measure (hence fractals). A sphere is the boundary of a ball; some neighborhood of every point is homeomorphic to an open ball, not sphere. Omitting the importance of transition maps ignores the huge world of differential manifolds, for which the very definition relies on properties of the transitions — the "stitching", if you will.

And although it does not appear in the lead, let me once again insist that calling Earth's surface "flat" is a bizarre departure from reality. You people need to get outside more and look around (for most parts of the planet). :-D --KSmrq^T 23:46, 26 April 2006 (UTC)

There is a big difference between "locally" and "at small scales". If I spelled out what "locally" means, it means "in a neighborhood of every point". On the other hand I can't imagine what "at small scales" means, given that the open interval (0, 1) is homeomorphic to the entire real line. The absence of scale in topolotical spaces is one of their essential characteristics. Even if we limit ourselves to metric spaces, any constant multiple of a metric is again a metric.

Septentrionalis's suggestion is fine, to replace "locally" by "in a neighborhood", since the technical definition of neighborhood is very close to the non-technical meaning.

And of course I meant n-ball. My bad.

I'll make those two changes above, and this time remember to sign my proposal. Rick Norwood 22:44, 26 April 2006 (UTC)

You're right about the phrase "at small scales". It's too vague to have any meaning. The current solution looks good to me. Now let me say that I'm vaguely uneasy with the heavy reliance on dimension in the introduction, though I'm having trouble putting my finger on exactly why. Instead of "manifolds are usually described by their dimension", how would it be with "Like Euclidean space, manifolds have a notion of dimension." And after the surfaces, some mention that manifolds allow for arbitrary (and let's not address whether "arbitrary" includes infinite) dimension. If the first paragraph mentions only some examples of curves and surfaces, the reader may be left with an impression that manifold is just an inclusive word for curves and surfaces. I dunno.. what do you think? -lethe ^{talk +} 23:35, 26 April 2006 (UTC)

My reasoning on "dimension" is as follows. We need to give some examples, and we need to make clear the difference between a manifold and a CW complex. That difference is, essentially, that all parts of a connected manifold have the same dimension. While people may not know what "dimension" means in general, I think most people know the difference between a line and a plane. If we avoid the word "dimension", how to we get across the idea that a manifold can't start as a plane and then thin out into a line?

Objections:

Neighbourhood of a point does not convey any idea to a layman. We need to use simpler terms, like very small area around the point (ignoring conditions like open).
What is an Euclidean Space? As the concept of Euclidean Space is so fundamental to the concept of a manifold, we should at least give the reader an idea that we are talking about your ordinary geometry rather than depend on him learning it from the article on Euclidean Space.
"Manifolds are usually described by their dimension." Not true. Dimension is a property of a manifold, specifying the dimension does not specify the manifold.
It misses a sentence important for understanding, "but which may have a more complicated structure when viewed as a whole". This may be obvious to the mathematician, but not to the layman.
The example part should be in the second para like it is now.
The intrinsic view of "gluing together" should be mentioned in the first para, like it is now. We should also refer to non-uniqueness of a coordinate choice, as mentioned by someone along the line.

I also take it that you are proposing to change only the first para and not the entire lead. Incidentally, who first raised an objection to the lead as it was then? Loom91 06:54, 27 April 2006 (UTC)

I can live with the current intro, but please streamline the grammar in the first sentence! I also worry that Euclidean in the first sentence gives a misleading impression, but again, I can live with it. How about one of the following

In mathematics, a manifold is an abstract mathematical space which has no distinguishing features when viewed up close, but which may have a complicated structure when viewed as a whole.

In mathematics, a manifold is an abstract mathematical space which, in a close-up view, resembles a space governed by elementary geometry, but which may have a more complicated structure when viewed as a whole.

User:MarSch's patchwork metaphor, mentioned above, is also a good first sentence.

Rmilson 13:24, 27 April 2006 (UTC)

As discussed above, I think the non-mathematical definition of neighborhood and the mathematical definition are close enough to convey understanding, whereas the idea of a "small area around a point" conveys the wrong impression, that is, that neighborhoods must be small.

Euclidean space is linked. I have no objection to explaining it here if (and this is why I didn't try) you can do so in no more than a sentence or two. If a person really has no idea what Euclidean space means, they can follow the link.

To say that manifolds are usually described by their dimension is not the same as suggesting that they are specified by their dimension. But any discussion of manifolds in mathematics usually begins by specifying the dimension. See, for example, 3-Manifolds by Hemple or Topology of 4-Manifolds by Freedman and Quinn.

I'll restore the mention of more complicated structure.

I'll put the examples in the second paragraph.

While some manifolds are constructed by "gluing", many are not.

I'm only working on the very beginning of the introduction because an attempt to do more was instantly reverted. However, I note that while I'm here trying to work with the other people interested in editing the article, many people have made major rewrites of the article without discussing them.

The problems with "close up" are discussed above. Objections to the "patchwork" idea are what started the current bout of revision. Rick Norwood 14:52, 27 April 2006 (UTC)

This discussion has rambled through so many suggestions and replies I'll just pick a spot and indentation to comment. First, this is explicitly a survey article. It must begin by speaking to a broad, general, untrained audience. It must include, not exclude, topological manifolds, differentiable manifolds, Banach manifolds, and so forth and so on. It must leave the details and implications of the specialities to the specialized articles. Such an article is surprisingly hard to write well, much harder than the specialty articles. The lead and introduction attract a disproportionate amount of attention, and thrashing rather than concensus. Every Johnny-come-lately editor (and I was once one) sees a problem and knows just how to fix it; and so does the next, and the next, and ….

Mathematicians see many definitions; precise postulates are our bread and butter. Yet as writers for the general public, it is clear that few mathematical editors appreciate the larger use of definitions, as covered by the Wikipedia definition article. Folks, it ain't easy! We would like someone who encounters the term "manifold" in who-knows-what context to be able to read the first paragraph and get the general idea. We must not depend on unfamiliar mathematical language, but rather on familiar common experience. We need not try to pack all the possibilities into a single sentence; in fact, that approach is doomed.

What should we mention about manifolds in the lead? I'd suggest we say something to indicate that our focus is not number theory or abstract algebra, but topology and/or geometry. We need to point out local uniformity (as a rule), global flexibility, and overlap consistency. This methodology is echoed in other constructs as well, including fiber bundles and schemes, but has important differences from the gluing of a CW complex.

And we need to realize that no matter how brilliantly we write, someone — now or later — will be dissatisfied. --KSmrq^T 21:14, 27 April 2006 (UTC)

I like your point of view about this article a lot. I'd like to see how that point of view translates into article space. Have you written a version of the intro that we can consider for inclusion? -lethe ^{talk +} 21:57, 27 April 2006 (UTC)

I'll wait for a few more comments, and then put the paragraph above in place of the one we've got now. Not everybody likes this one, but not everybody likes the old one, and while I try to establish a consensus, the article is rapidly changing.

On the subject of definition -- just how different the views of various people are on that subject was brought home to me when my son, a college student, reported the following exchange in an English class on critical thinking. He said something to the effect that a definition was arbitrary, and any definition would do, as long as everyone in the discussion agreed to observe it. The teacher was of a totally different view -- definitions are right or wrong, and if everyone agrees to a wrong definition, it is still wrong. I think you can guess which side of the argument I'm on. Rick Norwood 21:38, 27 April 2006 (UTC)

I think that prof's view is pretty much totally untenable. -lethe ^{talk +} 21:51, 27 April 2006 (UTC)

Several hours have gone by without an objection, so I'm going to replace the current intro. If you want changes, please move forward rather than back. Rick Norwood 00:28, 28 April 2006 (UTC)

I made one change in the intro above. I replaced "looks like" with "resembles" because it seemed to improve the flow. Rick Norwood 00:40, 28 April 2006 (UTC)

The role of Manifold in the cluster of manifold articles

At the moment there are articles Manifold, Topological manifold, Complex manifold, Differential manifold Symplectic manifold, Riemannian manifold, Pseudo-Riemannian manifold, Kähler manifold and Calabi-Yau manifold. Clearly the last eight are about precise mathematical concepts and this one is a more general article. Shouldn't the main focus here be (1) an intuitive overview and history (well-addressed at present) and (2) a comparison of the different concepts and their interrelationship. If so, is there an argument for rearranging material (and avoiding sections dealing with exactly the same things)? For example, this article defines a manifold using the topological manifold definition, which is both doing something that is done in the more specialised article and providing only part of the truth for the other types of manifold (like the relevance of the topology of the plane to Euclidean geometry). The general description of atlases is much more appropriate though, but could do with more emphasis on the fact that for each class of manifolds we demand different compatibility conditions. Any views? Elroch 15:55, 27 April 2006 (UTC)

What you propose sounds good to me. I wasn't aware of the article topological manifold. A quick look suggests that this article and that article should be merged. Rick Norwood 17:35, 27 April 2006 (UTC)

I may be exposing my biases here, but it's my opinion that if one of those articles were to be merged with this, it should be differential manifold, not topological manifold. -lethe ^{talk +} 20:04, 27 April 2006 (UTC)

No, a merger is wrong. This article was explicitly created and designed as an overview, while an article on topological manifolds needs to focus on features specific to topology, ignoring all the variant structures of, say, Milnor spheres. Similar remarks apply to each of the specializations. --KSmrq^T 20:13, 27 April 2006 (UTC)

Point taken. Still, there is a lot of repetition. Given that all those articles exist, we ought to use them for the more technical aspects of this long article. Rick Norwood 21:40, 27 April 2006 (UTC)

Yeah, no merger. We'd have to have a big fight to see whether topological manifolds are "the most important" class of manifolds or differential manifolds are. And what fun would that be? And of course, this article would have to become far more technical, whichever choice we made. On a related note, do you guys consider differential structures to be "geometric" or "topological"? I lean towards the latter, and consequently feel comfortable calling the the exterior derivative a concept of differential topology (rather than differential geometry), but I'd like to hear other people's thoughts (over at talk:exterior derivative, where rmilson brought it up). -lethe ^{talk +} 21:50, 27 April 2006 (UTC)

Between differentiable manifolds and topological manifolds, it is not a question of which is more important but which is simpler. As for whether differentiable structures are geometric or topological, I would say, neither. I'd go with analytic. This is how I divy up pure math. It all starts with numbers and shapes. Out of that comes algebra and geometry. Combine the two to get analytic geometry. Throw limits into the mix to get calculus. Extend algebra to abstract algebra, extend geometry to topology, extend calculus to real and complex analysis. Combine abstract algebra and topology to get algebraic topology. Combine abstract algebra and analysis to get Frechet spaces, Minkowski spaces, Hilbert spaces and the like. Combine topology and analysis to get differentiable manifolds. Put into a large bowl, toss well, and out comes category theory. Look at them through a microscope and find set theory. Logic is the dressing that keeps it all from falling apart. Rick Norwood 22:53, 27 April 2006 (UTC)

I disagree with a lot of what you say.

What the article manifold is about is dictated by our naming conventions. What is the most common/important usage of the word, and what is the least ambiguous. Those are the two rules for naming articles, which are sometimes in contradiction with each other. The first rule says this should be about differential manifolds (if you think that's the most common/important class of manifold, as I do). The second convention says that this article should be about all manifolds (since that's the least ambiguous interpretation of the word manifold). We have no naming convention which says that our article should be about the mathematical object with the smallest structure.
I think limits give you topology, not calculus. For calculus, you need derivatives or integrals (that is, you need a differential structure). Lots of places have limits which I would not call calculus
Extend geometry to topology? You think topology is an extension of geometry? Even though topological spaces have smaller structures than (many) geometric spaces? This seems to be in contradiction with, for example, your position on the naming. And in any case, I can't support viewing topology as an extension of geometry, just because you take the course in topology after you take the course in geometry.
Combining algebra and topology ought to give you topological groups, topological rings, etc. Algebraic topology is not really a combination of the two, even though you need material from both your topology course and algebra course.
Do you have a different definition of Minkowski space than I do? For me, Minkowski space is a finite dim vector space with a non-definite inner product. A decidedly geometric place, for doing relativity theory.
About Frechet space and Hilbert space, I divide my TVSes into abstract TVSes and concrete functional spaces. The functional spaces are what are studied in analysis (which is, after all, the study of real or complex numbers and functions). Lebesgue space, Hardy space, Sobolev space, etc
Therefore I view abstract TVSes like Hilbert space, Banach space, Frechet space as objects in the coincidence of algebra and topology (which, as I said, is not algebraic topology). They are topological groups. Of course, there is necessity to know about these guys in order to do analysis in an abstract way.

on the other hand, I might agree with some extended version of your description of category theory. I might also agree with your description of set theory if you had said "some choice of microscope". There are other choices. -lethe ^{talk +} 23:22, 27 April 2006 (UTC)

The sort of picture I had was to introduce some definitions which have a higher level of abstraction (eg an abstract notion of compatibility of charts which would, hopefully, be applicable to all the types of manifolds which a specific choice of the compatibility condition). Also, it would be nice to describe the tree of types of manifolds with increasing amounts of structure in a unified way, perhaps with a nice diagram. Another issue might be to cover the ways in which different types of manifold can be given more structure in ways which may, or may not, be unique to within isomorphism. This is rather a lot, but I am sure there are several who know parts of this picture very well. With regard to the difference between differential geometry and differential topology, perhaps it doesn't matter, as long as one agrees on the axioms. :-) Elroch 23:04, 27 April 2006 (UTC)

I was just riffing on the idea of structure, in response to your question of whether differentiable manifolds were geometric or topological. It's fun to talk about, but not relevent to this article. So, in the spirit of fun, I'll respond to your comments, after an

I agree that this is fun. A lot more fun that actually trying to write the article, that's hard work! This talk page is swamped with miles of comments which are mostly not helping the article and are pissing off KSmrq, so I feel a little bit bad about prompting this series of digressions. Oh well. -lethe ^{talk +} 00:58, 28 April 2006 (UTC)

((off topic)) warning to people with better things to do with their time. Historically, the application of limits to calculus and the application of calculus to limits came long before the invention of topology. Essentially, calculus is the study of three linear operators, the limit, the derivative, and the integral. Those operators were invented in the reverse of that order, but the limit comes first logically. Topology is an extension of geometry in the direction of abstraction. Both deal with sets of points, and many of the examples from topology arise from geometry, just as abstract algebra is an extension of algebra in the direction of (what else) abstraction. And many of the examples of abstract algebra rise from the real numbers. And, of course, out of the combination of topology and abstract algebra, you get both topological algebra and algebraic topology. Funny but true story: when I was doing the paperwork for my Ph.D. an administrator in the registrar's office didn't want to give me credit for both of those two course because to him they sounded the same.

((back on topic)) Anyway, the important point is this: the reason for starting with a simple structure is that this seems to be a portal page, and portals should be widely accessable. As for which kinds of manifolds are "most important", it seems clear to me that the answer is that everyone thinks the kind of manifolds he or she works with are most important. I'm in knot theory, work almost entirely with 3-manifolds, and almost never use differentiable structure. But I am aware of the importance of other kinds of manifolds. Rick Norwood 00:23, 28 April 2006 (UTC)

Well, as a portal page, I might be able to get on board your argument. Give some prominence to the (layman's description of the) topological manifold. But I shouldn't like to see other classes of manifolds fall too far behind in their consideration in this article. -lethe ^{talk +} 00:58, 28 April 2006 (UTC)

RN: Thanks to Elroch for putting back the links, which evidently "cut and paste" removes. And I agree with Lethe that differentiable manifolds, at least, need to be high in the article. Rick Norwood 12:28, 28 April 2006 (UTC)

Paragraphs three and four

Paragraph three seems fine to me as is.

I would like to rephrase paragraph four in paragraph form rather than in bullet point form. The Wiki style sheet does not forbid bullet points, but frowns on them. Here is how paragraph four reads now:

"Additional structures are often defined on manifolds. Examples of manifolds with additional structure include:-

differentiable manifolds on which one can do calculus
Riemannian manifolds on which distances and angles can be defined
symplectic manifolds which serve as the phase space in classical mechanics
the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity."

I suggest:

"Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity."

No change in the words (except the addition of "and") only in the punctuation.

Comments? Rick Norwood 00:45, 28 April 2006 (UTC)

Oh definitely. This should be not controversial, just go ahead and do it. I would probably support some cleaning up of the language too, what do you think? Perhaps conflating pseudo-Riemannian structures with Riemannian, and mentioning GR along with Riemann? -lethe ^{talk +} 00:48, 28 April 2006 (UTC)

I'm going to take your advice and do it, and you can fix the Riemannian stuff -- for me, Riemannian manifolds, Bozhe moi! . Rick Norwood 00:52, 28 April 2006 (UTC)

The wikipedia guidance against bullet points seems excessively pedantic. The justification for this guidance is that it follows normal practice in paper encyclopedias, where a bulleted list uses more paper. On a web page it has no such disadvantage and makes the separation between items in the list much clearer to the reader. I am strongly in favour of the use of bulleted lists rather than free format text, simply on the basis that they are definitely more effective at getting the information across to the reader. Elroch 11:51, 28 April 2006 (UTC)

Hmm.. I guess I was wrong about this being non-controversial. Well, anyway, I didn't like the list in this instance, although I'm not opposed to using bulleted lists in general (check out the article Locally convex topological vector space, which I wrote, where I use itemized lists throughout the entire article. I think I went overboard.) -lethe ^{talk +} 12:09, 28 April 2006 (UTC)

I agree that there is a place for bullet points, but that they are not needed in this paragraph. Rick Norwood 12:29, 28 April 2006 (UTC)

The Circle

I've been reading the next section, The Circle, in light of the following reviewer's comment:

"a lot of complex mathematical terminology is not even linked to, never mind explained."

It seems to me that, at least in this section, the links have been provided now, and the pictures are pretty, but I have to wonder who the section is intended to address. If the audicence has, say, a working knowledge of analytic geometry, then the exposition is elementary, but what will "projection onto the first co-ordinate" mean to a majority of readers?

not much; and it should be possible to explain the idea without the vocabulary. Septentrionalis 22:12, 28 April 2006 (UTC)

The article has to become technical at some point, but is this too soon?

Comments? Rick Norwood 01:14, 28 April 2006 (UTC)

I don't think it is possible to treat Manifolds without assuming at least Analytic Geometry after the lead. Analytic Geometry is high-school math, so we should be fine unless we are facing primary school students or adults who have managed to completely forget their high-school math, and I don't think we need to consider those readerships beyond the lead. Loom91 08:17, 28 April 2006 (UTC)

Many well known people have embraced the quote, "There is no such thing as algebra after high school," including people who should no better. Rick Norwood 12:31, 28 April 2006 (UTC)

The arcs of the circle are described as semicircles, but not drawn as such. shouldn't they each be 180 degrees, or am I missing something? Ojcit 19:13, 29 September 2006 (UTC)

Yes, you're right, the picture doesn't match the text. The yellow arc is certainly not the entire top half of the circle. Michael Kinyon 19:18, 29 September 2006 (UTC)

No, you're both wrong, though the mistake is natural and we've discussed the relationship between language and figure before. While it is true that the yellow arc itself is not a semicircle, its horizontal extent is exactly half the circle, as can be seen from the projection. That is, the yellow arc covers the top half of the circle, but obviously is not even part of the circle. If the yellow arc were actually drawn as a full semicircle, it would be huge, cumbersome, and less helpful. How do we know? We tried.

I would support rewording "(the yellow part in Figure 1)" to read "(indicated by the yellow arc in Figure 1)". On the other hand, from a topological point of view it is completely irrelevant whether we use the full top half or just enough to have an open set in common with its neighboring charts. Therefore we could refer to "arcs" rather than "semicircles".

Personally, I trust readers to be able to absorb the concept discussed and illustrated in this introductory section without the encumbrance of revised language. So far, my trust has been rewarded. --KSmrq^T 00:12, 30 September 2006 (UTC)

Well, in my case, you're preaching to the choir. I know what is and isn't relevant topologically, and I know that the yellow arc is not intended to be part of the circle. All I said is that the figure doesn't match the text, and indeed, it doesn't. It says that the yellow part is the upper half. Instead of "indicated" how about "(covered by the yellow arc in Figure 1)"? It's actually closer to what is meant while keeping the informal tone of the section.

Regarding the accuracy of the figure (which I concede is not as important as its pedagogical utility), it actually took me a few seconds of staring to see that the yellow and red arcs do, in fact, "line up" correctly with the circle. The reason I did not realize it at first is that it is immediate to the eye that the blue and green arcs do not line up correctly; they and their corresponding vertical line segments are too short. I would suspect that lengthening them slightly would be too difficult at this stage of the game.

Your trust, incidentally, is quite heartening. Michael Kinyon 12:34, 30 September 2006 (UTC)

I'm a little uncomfortable with "covers", partly because it has a technical meaning in topology (as in “covering map”), and partly because it works less well for the other charts.

As for trust, in the end we have no choice. Until we develop a technology that can transfer understanding directly from one brain to another, learning will remain an active process for the reader. We cannot do all their thinking for them, and it is counterproductive to try. Good writing does not just instruct readers, it engages them.

In fact, when I give a lecture I often like to leave an open question or two, to engage the audience. The kind of people who like to learn and do research find it irresistible. Likewise, a wise author ends a paper with suggestions for further research, partly because that will lead to more interest and more citations! (A famous example is Hilbert's problems.) --KSmrq^T 20:49, 30 September 2006 (UTC)

Good point about "covers". It's informally correct, but could be confused with its technical meaning. Maybe your suggested rewording is the way to go.

I don't think anyone would disagree with your platitude about what constitutes good writing. (Yes, I given open questions in talks and papers, too.) It's simply that in the case of the sentence over which we're splitting hairs, we have writing which is neither instructive nor engaging. The rest of the section is fine.

By the way, the hairs I'm splitting here are grey, not yellow, red, green, or blue. :-) Michael Kinyon 21:46, 30 September 2006 (UTC)

Conformal manifolds

Looking at the introduction again, it struck me that there is an obvious class of manifolds which isn't mentioned, namely those where angles are meaningful but distances aren't, which I would call "conformal manifolds" (it appears others agree, but this seems to be given only limited coverage on wikipedia, in the article Conformal geometry). In the 2-dimensional real case, these would be the Riemann surfaces, with holomorphic transition maps, but in higher real dimensions the transition maps would need to be higher dimensional conformal maps (just Mõbius maps, I think). Elroch 12:32, 28 April 2006 (UTC)

I'm familiar with conformal transformations on Riemannian manifolds, but what exactly is a conformal manifold? Can the notion of conformal be defined without a metric? Edit: looking at the article, I see that it's an equivalence class of metrics. Makes sense. -lethe ^{talk +} 12:55, 28 April 2006 (UTC)

I believe a conformal manifold can be defined directly by saying it is a manifold where the transition maps are conformal maps between open subsets of Rⁿ, as this will ensure angles are well-defined. The extra structure of tangent bundles on Riemannian manifolds is not really necessary, and is a bit like defining a topological manfold starting from a differentiable manifold.

But you need an (equivalence class of) inner product on Rⁿ to define conformal maps there, which means a (n equivalence class of) metrics on the manifold. -lethe ^{talk +} 14:06, 28 April 2006 (UTC)

On a little further reflection, the issue is whether the very limited conformal maps available for n>2 mean that there is some way of getting a unique metric structure on a conformal manifold, which would justify not treating them as a special class (except in dimension 2) Elroch 13:16, 28 April 2006 (UTC)

Yes, maybe that's the right way to think of it. A Riemannian manifold is a manifold with a reduction of the structure group of the frame bundle from GL(n) to O(n) while the conformal manifold ought to be a reduction to the conformal group. -lethe ^{talk +} 14:18, 28 April 2006 (UTC)

There's a couple of other ways to describe conformal structure. A common approach is to say that a conformal structure is an equivalence class of Riemannian structures: two metric tensors are judged equivalent if they differ by scaling factor. One can also employ a Cartan connection. Conformal structure is a well developed topic in differetnial geometry and is of considerable interest in GR. Kobayashi's Transformation Groups in Differential Geometry is a reference. Rmilson 14:27, 28 April 2006 (UTC)

Thanks, Rmilson. So basically, the metric (to within a scaling factor) comes "free" with a general conformal manifold, just like in the case of Riemann surfaces, despite the much smaller conformal group? Might be worth noting somewhere. Elroch 15:34, 28 April 2006 (UTC)

Of course the smaller group makes it easier, not harder, and the key thing is that all conformal maps are very nearly orthogonal maps near any point. Elroch 15:38, 28 April 2006 (UTC)

The comment about the metric coming 'almost for free' is spot on. The issue of the group is trickier, and I don't think the comments to date get it quite right. First of all, the idea of getting conformal structure by reduction of structure of the frame bundle doesn't work. The problem is that the group of conformal transformations of 'n-dimensional space' is larger than GL_n, not a subgroup of GL_n. Likewise conformal n-space is not just R^n, but rather R^n + one point at infinity --- topologically S^n. Usually the way one gets CO(n) (the conformal group) acting on conformal space is to have the proper Lorentz group, the connected component of SO(n+1,1), act on the projectivized light cone, PK^n. To be precise, we start with SO(n+1,1) acting on n+2 dimensional Minkowski space, then we restrict to the action on the n+1 dimensional lightcone K, and then we projectivize to get the action of SO(n+1,1) on the space of null lines, thats PK^n. For example, CO(2) is a subgroup of SO(3,1), and 2-dimensional conformal space is S^2, the celestial sphere of past (or future or both) directions. So one can't do conformal geometry by using G-structures. One really needs a Cartan connection, which is a slightly different gadget.

Ah. Well, not larger than GL(n) (there are linear maps which are not conformal), but not contained in GL(n) either (since there are conformal maps which are not linear). So we shouldn't expect to get a vector bundle at all. Thank you, that cleared up some of these things a bit for me. -lethe ^{talk +} 22:13, 28 April 2006 (UTC)

A final remark about a potential pitfall in all this. I think there was some mention of the Liouville theorem to the effect that conformal transformations (when one considers such things locally or infinitesimally) of n-dimensional space for n>2 form a finite-dimensional Lie group. However the relationship between this group and a general (curved) conformal structure is rather subtle. It's analogous to the relationship between Euclidean transformations and Riemannian geometry. Sharpe's book on Cartan geometries is a reference. Cartan's own writing is also a great resource. The notation is archaic, but he had a certain eloquence and was good at communicating geometric ideas and motivation. A useful resource in this regard is here [1] Rmilson 17:10, 28 April 2006 (UTC)

It's probably worth noting that to define a conformal manifold in the direct way I mentioned, the transition maps are between open subsets of Rⁿ, so there's no need to deal with the 1-point compactification. It's my understanding that just like in differential geometry, one can construct the manifold using an atlas, from which all the bundles arise naturally, as a convenient alternative to defining the manifold using bundles. (I still have Graham Allen's notes on this somewhere around). With regard to Lethe's comment about the metric coming from the inner product on the open subsets of Rⁿ, this isn't quite trivial, as conformal maps do not preserve distances. Very close to a point, however, they almost do, which allows the extension of a concept of distance from near a point to the whole space, intuitively by tiling it with very small tiles on which distances are well-defined. Elroch 20:04, 28 April 2006 (UTC)

P.S. it's not trivial that this is well defined when you get to a point using two different routes, but it can be proved for Riemann surfaces: anyone know this result for a higher dimension conformal manifolds? Elroch 20:08, 28 April 2006 (UTC)

I believe that the construction you describe will yield the trivial conformal geometry. In 2D, the group of local conformal transformation is much richer (holomorphic functions) than the group of global conformal transformations (Mobius xforms). In 3d and higher, all local conformal transformations are restrictions of global ones -- that's Liouville's theorem. Rmilson 20:45, 28 April 2006 (UTC)

Well that's what comes of extrapolating what I half recall about Riemann surfaces and differential geometry with a bit of guesswork. I should defnitely brush up on this stuff. Elroch 22:02, 28 April 2006 (UTC)