Talk:Riemann surface

Early 2003

How about an explanation in layman's english, too? Theanthrope 06:46 Jan 23, 2003 (UTC)

I'll try. However, while it's possible to sort of talk about a manifold in layman's english, the distinction between Riemann surfaces and 2-manifolds is very subtle and difficult to describe without using any math. Loisel 07:08 Jan 23, 2003 (UTC)

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Axel, I saw you improved the opening paragraph. However, I'm concerned that perhaps we could serve Theanthrope's request (see above) by moving some of the jargon to a second paragraph, and keeping the opening paragraph as light on jargon as possible. I think it would also help if someone could give an interesting example of a 2-manifold that can't be viewed as a Riemann surface. This way, we can somehow say "Riemann surfaces are mainly what everyone thinks of as surfaces, however some surfaces such as X aren't Riemann surface, see below for technical details." Unfortunately, I'm not sure which 2 surface does not admit a Riemann surface structure. What do you think? Loisel 06:18 Feb 12, 2003 (UTC)

I moved the term "holomorphic" into the second paragraph, even though I think it's the whole point of the enterprise. We certainly need examples: open sets of the complex plane, the Riemann sphere, the two-sheet thingy you get from the square root function etc.

Everything2 has a nice writeup at http://everything2.com/index.pl?node_id=1007348

All Riemann surfaces are orientable, so the Moebius strip can't be made into a Riemann surface. AxelBoldt 16:56 Feb 12, 2003 (UTC)

I didn't know that. I added an argument outline to the article. I didn't want to infect the article with a lengthy proof. Delete it if you don't like it. Loisel 03:27 Feb 13, 2003 (UTC)

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We need to de-TeX this article a bit, it takes far too long to load. -- Tarquin 16:39 Feb 12, 2003 (UTC)

Yes, I agree. AxelBoldt 17:15 Feb 12, 2003 (UTC)

Me too - plus it's horrible to read text where the font size jumps up and down by a factor of 2. Chas zzz brown 12:04 Feb 13, 2003 (UTC)

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Axel, I'm not sure I have the correct idea for the charts of concrete riemann surfaces. I'm not completely sure that what I have in mind is a chart independent of the a_k. Can you clarify that for me? Loisel 22:07 Feb 17, 2003 (UTC)

Actually, now that I think about it: it doesn't seem like all "concrete Riemann surfaces" as I defined them are Riemann surfaces. It seems the concrete ones may have singularities while our abstrace Riemann surfaces cannot. I don't know how to fix this right now; maybe we should just take the example of concrete surfaces out, even though historically it was the motivating one. AxelBoldt 00:21 Feb 18, 2003 (UTC)

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I'm interested in the "only if" portion of what you just wrote. It says that if a real two-manifold X is orientable, then it has a Riemann surface structure. Is there an easy way to see this, or a book where I can look this stuff up? I've been sitting here trying to take some charts and making sure they're Riemann compatible, but try as I might, I can't figure it out. Take for instance R^2. In some open region, I use the identity map for a chart f(x)=x. In another open region, I use a function g(x) which isn't analytic at some point (like exp(-1/x^2) at 0 or something like that.) How do I turn these charts into Riemann charts? Do I need to know some facts about the ring C^\infty(X) to prove this? Loisel 02:32 Feb 18, 2003 (UTC)

I found the claim in http://www.jchl.co.uk/maths/Riemann%20Surfaces.pdf I don't know how to prove it. But I do know that every differentiable real manifold has an atlas whose elements are compatible as real analytic functions. You somehow need to use the orientability to get from real analytic to holomorphic. AxelBoldt 06:23 Feb 18, 2003 (UTC)

Can you point me to the page you're thinking of? Are you referring to the realization of Riemann surfaces as algebraic surfaces of the form you suggested ("concrete Riemann surfaces") or is it something else? Those are good notes, by the way. Loisel 07:34 Feb 18, 2003 (UTC)

On the top of page 6 of the above given PDF file, he makes the claim that every orientable real 2-manifold can be turned into a Riemann surface. I don't know if he proves it later in the notes. AxelBoldt 16:56 Feb 18, 2003 (UTC)

I see. Interesting. Loisel 17:50 Feb 18, 2003 (UTC)

sheaves

Has anyone else seen the definition of "sheaf" used in this article. I was under the impression a sheaf was a contravariant functor from the category of open sets and inclusions in a topological space to the category of sets, rings, etc. such that if all restrictions of certain sections line up a unique new element can be patched together from the data. The "sheaf" in the article seems to be a global analytic function constructed as the union of the functions given by local power series expansions; it seems like this object would be a section of some sheaf of holomorphic functions rather than a sheaf itself. Is this wrong? Vivacissamamente

The usage isn't exactly correct, perhaps; but it is rather close. That is, with the definition of sheaf space of a sheaf (an alternative to the functor approach, and earlier too), the projection of G really will be the kind of local homeomorphism that is to be identified with a sheaf. As if the Riemann surface required is really constructed as a quite small subsheaf of the sheaf of holomorphic functions, would say it another way.

Charles Matthews 14:32, 8 Jun 2004 (UTC)

I've never been able to parse the algebraic sheaves, although I realize they're much more talked about than the complex analysis sheaves today. However, I believe that the nomenclature originated in complex analysis, and I'm fairly certain that the definition here is the one for complex analysis.

Loisel 23:12, 9 Jun 2004 (UTC)

Historically, I believe, this goes back to the Poincaré-Volterra theorem? Charles Matthews 07:23, 10 Jun 2004 (UTC)

Which one is that? Loisel 21:01, 10 Jun 2004 (UTC)

Nevermind, got it. Loisel 16:06, 11 Jun 2004 (UTC)

Analytic continuation

Would it make sense to move the whole bit about analytic continuation and sheaves to the article on analytic continuation? Its not clear to me why this stuff about analytic continuation is even in this article... (other that that g composed with f^-1 needs to be analytic, etc. Anyone up for that, or will my edits get reverted ??linas 20:03, 22 Jan 2005 (UTC)

Still planing on making good on the above threat...real sooner now. linas 14:59, 14 Apr 2005 (UTC)

The connection is because the 'maximum analytic continuation' is to a Riemann surface; and this is one of the historical roots of the topic. That said, it might be just as clear to place this material in the other article. Charles Matthews 15:44, 14 Apr 2005 (UTC)

Moved. I want to expand this article a bit, and the stuff on analytic continuation was taking up space. I also remember learning analysis for the first time, and being frustrated by the lack of a good definition of analytic continuation. For whatever reason, courses in begining complex analysis don't really use the words riemann surface very much. linas 17:14, 16 Apr 2005 (UTC)

sheaves

the article on sheaves contains a link to this article for an example of a sheaf. I removed this line from the text, because there is no such example here. If anyone does add an example at some point, then the link should be restored in the article on sheaves. -Lethe | Talk 19:53, May 31, 2005 (UTC)

Looks like the relevant material was moved in April to analytic continuation. Charles Matthews 20:48, 31 May 2005 (UTC)

Hyperbolic surfaces have nontrivial π₁?

the article says the following things: the disk is among the simply connected complex surfaces, the disk is hyperbolic, hyperbolic surfaces have nontrivial fundamental group. One of these three sentences doesn't fit. I think it's probably the last one? -Lethe | Talk 13:29, Jun 16, 2005 (UTC)

thinking more on the matter, I think a surface of constant negative curvature must have negative euler characteristic, and so high genus, and so nontrivial pi1. So the first statement is in error? Also, can I infer that the Poincaré half plane is multiply connected? That seems surprising. -Lethe | Talk 14:19, Jun 16, 2005 (UTC)

The last statement is not correct, the disk is the unique (up to biholomorphism) simply connected hyperbolic surface. The hyperbolic surfaces which aren't simply connected have the disk as the universal cover. The upper half plane is biholomorphic to the disk and so also hyperbolic and simply connected. -- Fropuff 15:05, 16 Jun 2005 (UTC)

Maybe the author of that stuff meant compact hyperbolic Riemann surfaces? The relationship between curvature and genus is maybe not straightforward for noncompact surfaces. I find similar erroneous statements in several other related articles as well. -Lethe | Talk 22:28, Jun 16, 2005 (UTC)

That is entirely possible. Statements further down (regarding the area of the surface) certainly apply only in the compact case. These statements all need to be clarified -- Fropuff 04:59, 17 Jun 2005 (UTC)

corrections

Yes, certainly it should be changed. In Riemann surface, it seems to me that the sentence

"Hyperbolic Riemann surfaces are multiply connected and thus have a non-trivial fundamental group."

ought not be the first sentence in the paragraph. I mean, this isn't particular to hyperbolic geometry, right? C is simply connected noncompact and parabolic, and C mod some rank 2 discrete lattice group is closed parabolic, which is not simply connected (pi1 = lattice group). At this point, I don't see what's special about the hyperbolic case: H is simply connected noncompact and hyperbolic. while H mod Fuchsian group is closed hyperbolic (pi1 = Fuchsian group). It's the same, so why is the hyperbolic case getting singled out?

I propose we change it to something along these lines:

"For every closed parabolic Riemann surface, the fundamental group is isomorphic to a rank 2 lattice group, and thus the surface can be constructed as C/Γ, where C is the complex plane and Γ is the lattice group. The representatives of the cosets are called fundamental cells. The area of such a surface is arbitrary (is this true?).

OK I copied this paragraph into the article. Notice the red links. Sould we redirect fundamental cell to free regular set for now, until someone writes a "real" article, or is that a bad idea? Should we redirect lattice group to lattice (group)? Or maybe redirect to torus? As to area of surface being arbitrary, that sure sounds right to me, but as you notice I'm error-prone. I left the sentance out. linas 00:33, 18 Jun 2005 (UTC)

Actually, there's already an article fundamental domain. I guess fundamental cell should redirect to that. And yeah, I guess lattice group should redirect to lattice (group). As for the area of a torus, the complex structure depends on the ratio of the lengths of the sides, so you can scale the area to anything you want. You can also see that the Gauss-Bonnet theorem can't tell you the area of the torus of constant curvature, since the curvature is zero. -02:10, Jun 19, 2005 (UTC)

"Similarly for every hyperbolic Riemann surface, the fundamental group is isomorphic to a Fuchsian group, and thus the surface can be modelled by a Fuchsian model H/? where H is the upper half plane and ? is the Fuchsian group. The representatives of the cosets of H/? are called fundamental polygons. The total area of such a surface , where g is the genus of the surface; the area is obtained by applying the Gauss-Bonnet theorem to the area of the fundamental polygon."

Can we say some more about the elliptic case? Is the Riemann sphere the only closed elliptic surface (up to biholomorphism)? We should link also to some article on Lie groups that says pi1 of universal cover mod discrete subgroup = discrete subgroup.

Last I looked, Riemann sphere was unique. As to the modulo-group remark, I think that the article covering map states this, if rather densely. Also, by lattice group do you mean wallpaper group? Or is there another definition? linas 14:03, 17 Jun 2005 (UTC)

No, a lattice group is not the same thing as a wallpaper group, although I think the latter includes the former as a special case. see Lattice (group). -Lethe | Talk 23:20, Jun 17, 2005 (UTC)

OK, so let me see if I have it straight: there is only one elliptic Riemann surface, and it's simply connected. For the parabolic case, there is one simply connected surface C, one surface with pi1=Z (C mod lattice of rank 1 = C*), and an infinite number of tori with pi1=ZxZ (C mod lattice of rank 2). And in the hyperbolic case, I'm a bit fuzzy. Is there only one surface for each genus >1? each of them has H as universal cover? -Lethe | Talk 02:10, Jun 19, 2005 (UTC)

Hmm. You know, I am at best a novice w.r.t Riemann surfaces. I don't know the parabolic case. Saying "its a lattice" essentially means "its a torus"; I have no clue if there are any parabolic riemann surfaces that aren't toruses. Can't say I've heard of such beasts, but also can't say I've heard of a theorem stating otherwise. Maybe its trivial? linas 00:33, 18 Jun 2005 (UTC)

Well, if the lattice group is rank 1, you get a parabolic manifold that is not a torus. You get the cylinder, which is the same as the multiplicative group of nonzero complex numbers. C, C*, and a bunch of tori, those are the only parabolic surfaces. -Lethe | Talk 02:10, Jun 19, 2005 (UTC)

The article on hyperbolic geometry has the similar statement "Hyperbolic surfaces have a non-trivial fundamental group , known as the Fuchsian group. The quotient space H/? of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface.", which also needs to me changed. I thought I saw something like this in a third article too, this morning, but now I can't find it. Maybe it was just the two. -Lethe | Talk 05:42, Jun 17, 2005 (UTC)

Several quick remarks: don't sacrifice overall clarity of an introduction to exactness. Most hyperbolic surfaces do have non-trivial genus; only one doesn't, the hyperbolic disk. I think that a minor edit could take care of the one-exception case. Other than that, the above proposed paragraphs look reasonable. linas 13:51, 17 Jun 2005 (UTC)

Sure, most manifolds in any isogeny class have nontrivial pi1, except for the one that doesn't. This is true for any isogeny class of Lie groups. But what does that have to do with hyperbolic geometry? The sentence makes it seem like somehow the hyperbolicity implies nontrivial pi1. That's true in the compact case, not true in the noncompact case. -Lethe | Talk 22:36, Jun 17, 2005 (UTC)

My apologies, I just realized you were complaining about sentences I had written back when. I am sorry they're misleading, this was not my intent, and perhaps shows my ignorance of the subject. The corrections you propose seem reasonable; I invite you to make them, or you can wait for me to do it. However, I have a backlog of unfinished projects, so waiting my prove to be irritating. linas 23:37, 17 Jun 2005 (UTC)

Never mind, I fixed by copying you wording. Please review for additional corrections. linas 00:33, 18 Jun 2005 (UTC)

Also, is it correct to say the representatives are the fundamental polygon? Isn't a representative of a coset simply a point? -Lethe | Talk 05:54, Jun 17, 2005 (UTC)

No, its not a point. Technically, the representatives are free regular sets which can be converted to fundamental polygons after a set of operations that re-arrange the area of the thing so that its bounded by geodesics, etc. linas 13:51, 17 Jun 2005 (UTC)

I think a certain set of representatives form the fundamental domain/polygon/free regular set, but the representatives themselves are simply points. I mean, I get what it's trying to say, but I think as it stands it is worded incorrectly. Actually, you can see the article on free regular sets to see the correct wording. -Lethe | Talk 22:36, Jun 17, 2005 (UTC)

Oh gosh, very sorry, yes, I misunderstood, you are absolutely right; the wording in the article is wrong. What can I say? Brain in neutral, keyboard in overdrive. I'll fix this in a few minutes. linas 23:37, 17 Jun 2005 (UTC)

Hyperbolic/parabolic/elliptic terminology

I am a little confused as to the meaning of hyperbolic/parabolic/elliptic in regards to Riemann surfaces. This article defines a Riemann surface to be hyperbolic/parabolic/elliptic according to whether the universal cover is the unit disk, the complex plane, or the Riemann sphere. Or, equivalently, whether the surface carries a Riemannian metric with constant curvature −1/0/+1. However, the book by Farkas and Kra (Riemann Surfaces 2nd ed. 1992, ISBN 0-387-97703-1) uses a different definition for these terms. In section IV.3.2 they define a Riemann surface to be

• elliptic iff it is compact,
• parabolic iff it is noncompact and does not carry a negative nonconstant subharmonic function,
• hyperbolic iff it does carry a negative nonconstant subharmonic function (necessarily noncompact).

According to this classification the three simply connected surfaces are still hyperbolic/parabolic/elliptic as before, but the meaning changes for another surfaces. For example, all compact surfaces (of any genus) are elliptic according to Farkas and Kra but most (g ≥ 2) are hyperbolic according to this article.

Can anyone comment of the different definitions? Are either or both standard? Should we mention the Farkas and Kra meaning in this article? I note that the Farkas and Kra definition is potentially more interesting since nearly every surface is hyperbolic in the sense of this article (the sole exceptions being the Riemann sphere, the complex plane, the punctured plane, and complex tori).

-- Fropuff 8 July 2005 04:30 (UTC)

I don't know. Jost, Compact Riemann Surfaces doesn't appear to attempt such a naming convention. linas 15:29, 11 July 2005 (UTC)

No, I don't suppose Jost would, as by the title of his book he only treats the compact case, which are all elliptic according to Farkas and Kra. I'll have to see if I can find any other references. -- Fropuff 16:01, 11 July 2005 (UTC)

I've checked a few other sources. Both Ahlfors (Riemann Surfaces, 1960) and Beardon (A Primer on Riemann Surfaces, 1984) use the Farkas and Kra meaning for parabolic and hyperbolic. They don't define elliptic spaces, refering to those spaces simply as compact surfaces. These books also mention that hyperbolic surfaces (in their sense) are precisely those surfaces which admit Green's functions.

On the other hand, Forster's book (Lectures on Riemann Surfaces, 1981) uses the meaning described in this article. He notes however that no one really refers to the Riemann sphere as an elliptic Riemann surface, as it might cause confusion with elliptic curves (i.e. complex tori, compact surfaces of genus 1).

Anyone have any others? -- Fropuff 19:20, 11 July 2005 (UTC)

Gamelin's Complex Analysis has a chapter on Riemann surfaces, with no mention of elliptic, parabolic, or hyperbolic surfaces. Lang's Complex Analysis, doesn't do Riemann surfaces, nor does Wells' Differential Analysis on Complex Manifolds, nor does Kodaira's Complex Manifolds and Deformations of Complex Structures.

On the other hand, it is easy to find the words elliptic/hyperbolic/parabolic in a book on Riemannian geometry where they mean positive/negative/zero curvature. I guess that usage is the source of the terms in this article.

It would be nice if the two usages agreed, but I guess they don't. I guess they agree one which ones are the simply connected versions: the sphere is elliptic, the plane parabolic, the half-plane hyperbolic. But with the nontrivial surfaces, they seem to be in total disagreement. Farkas and Kra want the many-handled torus to be elliptic, while its Riemannian geometry is hyperbolic. Hrmm...

I guess I think the article should be brought in line with Farkas and Kra, but it's not enough to just shuffle the words, some stuff has to be said about subharmonic functions and such. Anyway, I always thought it was weird how, according to this article, there is only one elliptic surface. Hardly seems worth giving it a name, if there's only the one, right? -Lethe | Talk 22:35, July 11, 2005 (UTC)

Hi! I would stick with the curvature definition. I guess the different terminology comes from the different applications. Riemann surfaces play a role in many different areas. From the geometric viewpoint, you should name them after their curvature. sphere with curvature 1, torus curvature 0, the two special cases kind of, and then all higher genus surfaces with curvature -1 and the hyperbolic space as universal cover.

Note that if you care for any geo-"metric" aspects (maybe in contrast to purely analytic aspects) this seems very important, as the sphere is the prototype of the sphere (somewhat stupid) the plane a prototype for a neighborhood on the torus and a higher genus surface looks like hyperbolic space IH locally.

Hope that makes sense! Norman - nhilbert.de

formal definition

Hello,

I must say the definition given here looks the most accessible one I have found on the internet. However there is one thing I don't understand and perhaps someone could clarify this : when talking about compatible maps f and g, suppose f is a map from G (open subset in X) to H (subset of C) and g is analogously from I to J

we require that G and I are not disjoint( intersecting domains) but what kind of map is f after g^(-1) then

from J to H or are there more restrictions?

Yes, there's a restriction, its defined only on the intersection of H and I. The article Atlas (topology) is supposed to explain this in detail, but it doesn't  :( linas 22:09, 15 August 2005 (UTC)

I don't see why we need the appeal to Zorn's Lemma. To get a maximal atlas containing a given one can't we just grab all charts compatible with all the charts already in the atlas. Any pair of these new charts must be compatible because they're compatible with all the charts of the original atlas, which covers. Can someone back me up or smack me down? Amling 16:40, 22 January 2006 (UTC)

Sounds right. In the smooth category, your statement is Lemma 1.10(a) in Lee's Introduction to Smooth Manifolds, GTM. Melchoir 18:30, 22 January 2006 (UTC)

Brave New World

Just thought I'd note that in the book "Brave New World", they play tennis on what is described as Riemann Surfaces. :-) In case anyone wanted to know. —Preceding unsigned comment added by 62.31.86.15 (talk • contribs) 23:25, 1 January 2006

Dang, I don't have a copy of the book on me. Could someone else add it to the article? Melchoir 00:42, 2 January 2006 (UTC)

Never mind, I got it. Melchoir 00:48, 2 January 2006 (UTC)

Fundamental Error

The article makes this claim: " A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable."

Alas, this is simply not true. Perhaps it goes without saying that a Riemann surface must be Hausdorff, but this is not sufficient. It must also have a countable base to its topology. One counterexample is the square of the long line. Another quite surprising example is the Prüfer manifold: Start with an open half-plane H = {(x,y) in R^2  : y > 0}. For each (c,0) on the x-axis, we augment H in a strange way: Define H_c as the homeomorphic image of H by the (real analytic) homeomophism h_c: H -> H (which is strictly into):

                                            h_c(x,y)  =  (x,y) + (x-c,y)/sqrt((x-c)^2 + y^2).  

(That is, we push each (x,y) directly away from (c,0) by exactly one unit.) Now take the union of H_c with the original H, which has the effect of adding a semicircle of radius 1 about (c,0) to what was H (which for each c we identify with H_c). Finally, perform this augmentation once for each (c,0) on the x-axis. The result is a real-analytic surface, the "Prüfer manifold", which obviously does not have a countable base to its topology (or a countable dense subset), since we have added uncountably many disjoint (2D) semicircles to what was H. Such a manifold cannot be made into a Riemann surface.Daqu 06:10, 16 May 2006 (UTC)

"Manifolds", in the sense of this article, are implicitly second-countable. See Manifold. Melchoir 06:31, 16 May 2006 (UTC)

Granted, but this only shows that Wikipedia's various definitions of a manifold need to be put on a sounder logical footing. E.g., the entry Topological manifold makes no such assumption of Hausdorff or separability -- which is exactly as it should be.) When topologists get serious, they specify a manifold as being topological with such-and-such precise additonal structure. In the present case, a significant episode in the history of Riemann surfaces was to determine whether or not all Hausdorff locally Euclidean connected 2-manifolds with holomorphic transition maps had to be separable (or equivalently are second countable). The Prüfer manifold especially called this into question. The fact that a 1-dimensional Hausdorff connected complex manifold is automatically separable is quite surprising.

Perhaps just as surprising is that higher-dimensional complex manifolds defined only as locally Euclidean, Hausdorff, connected and having an atlas with holomorphic transition functions need not be second countable! (Just take the complexification of the Prüfer manifold.)

So while I don't advocate getting too technical too quickly, it's also a good idea not to make easily-misinterpreted statements either (like that every connected surface can be given a Riemann surface structure).Daqu 14:30, 16 May 2006 (UTC)

Actually, when topologists get serious, they do geometry ;) Sam nead 23:16, 23 October 2006 (UTC)

Attempting to understand

I've got a question which doesn't seem to be in the article. What are the conditions for a real 2-manifold to be equivalent to a Riemann surface? From what I gather they are that: if on any overlap between a (x,y) coordinate patch and a (X,Y) coordinate patch $Z=X+iY\,$ is holomorphic in $z=x+iy\,$; and that the Cauchy-Riemann equations hold: ${\frac {\delta X}{\delta x}}={\frac {\delta Y}{\delta y}},{\frac {\delta X}{\delta y}}=-{\frac {\delta Y}{\delta x}}$. Are those all the conditions? —porges_(talk) 10:52, 10 July 2006 (UTC)

Also, is this equivalent? "A complex function f(x + iy) = u + iv is holomorphic if and only if it satisfies the Cauchy-Riemann equations and u and v have continuous first partial derivatives with respect to x and y."

I guess the shorter version is "if a real surface has holomorphic transition functions and it satisfies the Cauchy-Riemann equations, is it the same as a Riemann surface?" —porges_(talk) 00:24, 12 July 2006 (UTC)

The first part is correct. If a surface has holomorphic transition functions, then it is a Riemann surface. Holomorphicity means that the transition functions satisfy the Cauchy-Riemann equations, so you don't have to say that twice. -lethe ^{talk +} 00:58, 12 July 2006 (UTC)

Would a simpler example be useful?

I'm adding content to the article complex plane, and I'd like to tie that article into this one. I think the original idea of a Riemann surface arose from the consideration of multi-valued complex functions, and that it's natural to associate certain simple Riemann surfaces with functions like z^⅔, or logz.

This article is very nice as it stands. But I fear it may be inaccessible to the general reader, who probably has an intuitive notion of what a Hausdorff space is, but who may be scared-orf by such big words.

Anyway, I'm thinking this article might be improved by re-ordering the sections just a little. Specifically, I'd like to move the "History" section ahead of the "Formal definition" and add one or two naive examples to this section. I think the tie-in between the Riemann surface for logz and the residue theorem is particularly instructive. One need not be a topologist to comprehend that picture.

Does that sound like a reasonable plan? I'm asking because I think there's a real tension between rigor and readability in many Wikipedia articles about math. My own inclination is to lean toward less rigor, at least in the introductory sections of an article. But I don't want to offend the mathematicians who think that rigor is everything. DavidCBryant 12:00, 2 January 2007 (UTC)

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What are you trying to do?

Altouugh I'm not a mathematician, I'm not illiterate in math either. I want to ask a rather difficult question about this page. It deals with how to write for the public and raises the issue of what purposes Wikipedia has and, in specific, the purposes of this entry.

If the authors of this entry are writing for each other, and for senior math students, graduate students, or their professors, then you must abandon the idea that Wikipedia is a **general** encyclopedia for readers on the Web. If you are writing for the general public, then you have to abandon the abstract approach you have adopted. The reason is that no one (other than yourselves) knows what you are talking about -- or cares. If that's your goal, well, accept your elitism and redefine the purposes of Wikipedia.

I'll give an example. Let's start with the high school idea of graphing a function by plotting points on paper. Like y = x^2. OK, we get some points, and everyone can see you get a curve if you have enough points. Now let's jump to complex variable: w = z^2, where w = u+iv and z - x+iy. With some patience, even a high school student can understand that: the function z^2 takes PAIRS of points on one plane and MAPS them onto another. Now, can you use this example to describe what a Riemann surface is?

If you can, then you'll be able to pass your graduate student qualifying examination in complex variable. If you can't, then no matter how much you know about Hausdorff, you do not yet understand a Riemann surface. The best reference is Felix Klein's elementary mathematics from an advanced standpoint, because it was THAT kind of question that Klein raised **and answered**. Another excellent source is Klein's book on Riemann surfaces and electrical fields.

The reason this question is so extremely difficult is that it requires thought and understanding, not the ability to use jargon. Understanding _precedes_ jargon. So my advice is to scrap the entire article and start from scratch, with the basic ideas you want to explore. That you have not done, and as a result -- and as long as Wikipedia is an encyclopedia for the general public -- this article is a failure.

Timothy Perper Timothy Perper 09:42, 22 August 2007 (UTC)

That seems a good suggestion. Please go ahead and improve the article. -- Jitse Niesen (talk) 11:36, 22 August 2007 (UTC)

Thanks, Jitse Niesen and David Bryant -- we seem to see eye-to-eye that some changes in content and exposition are needed in this article.

But it's not a matter, I think, of merely making a few changes here and there. I think the article needs to be recast much more thoroughly. Mathematically, it needs more and/or clearer material on analytical continuation, on single- and multiple-valued functions, on genus, and (especially) on branch lines, branch cuts, branch points, and singularities.

Moreover, we -- anyone -- aren't "doing" mathematics here. The article starts like any good academic (or textbook) exposition in math, with a formal definition followed by an elaboration, also formal, of various ideas. But the readers aren't mathematicians. They're intelligent members of the general public: this is Wikipedia, not a seminar in complex variable. So the article needs illustrations, examples, and development of ideas historically *and* intellectually. I like the idea of starting with drawing graphs and getting from there to mapping and from there directly to how we can map a multi-valued complex function -- which leads *directly* to the Riemann surface. Only later, with an impulse to abstraction did these initially quite visual and intuitive ideas become abstractions. But that is only a personal preference.

I'm certainly not the one to say *unilaterally* what the directions should be for this article. I am therefore reluctant to merely make changes in the name of "improvement." We need some consensus first, especially from the people who did the hard work of writing the article.Timothy Perper 13:42, 22 August 2007 (UTC)

The image

I'd rather not correct this myself, since I'm neither a complex analysis expert, nor a Wikipedia expert. This is just a thought:

The image of the surface for ${\sqrt {z}}$ depicted in the upper left corner is actually only depicting the real part of a multi-valued function $f:C\rightarrow C$. What about including also the imaginary part as a separate image? It might be confusing to beginners, that's for sure. Maybe it wouldn't be bad to also include a page about ways how to imagine such mappings and provide a link. I.e. functions $R\rightarrow R^{2}$ and $R^{2}\rightarrow R$ can be visualised simply in 3D, but that's about it. Even though 4D cannot be imagined, we can still get the general idea by emplying projections into 3D.

On the other hand, Wikipedia is supposed to be a reference rather than a textbook, so I don't know how appropriate this would be.

193.85.150.3 (talk) 11:27, 10 December 2007 (UTC)

Images of Riemann surfaces labeled as for for z^-2, z^-3, z^-4

These are depicted as having branch points at the origin . . . and without having poles there.

But the functions z^-n for n = 2,3,4 have no branch points, and each has a pole at the origin. Can it be that the images are in fact the Riemann surfaces for z^1/n for n = 2,3,4 ? In any case, something seems to need fixing here.Daqu (talk) 18:01, 9 February 2009 (UTC)

Riemannian manifolds?

This has been in the article for a long time, so perhaps I should think longer before writing this, but anyway: How is a Riemann surface supposed to be a Riemannian manifold as in the second definition?

Certainly the complex structure allows the measurements of angles and hence induces a conformal structure, but a Riemannian metric can also measure lengths. This we cannot do with a complex structure, as maps such as z -> 2*z are holomorphic, but not length-preserving. Right?

Regardless of this, I think that the unsourced phrase in the second definition hence the name Riemann surface is highly doubtful as it stands. -- Momotaro (talk) 20:56, 4 May 2009 (UTC)

It is correct. I've provided a reference. Jakob.scholbach (talk) 19:31, 5 May 2009 (UTC)

I'm not convinced ... This chapter shows that when you have a Riemannian metric on a surface, you can find a conformal structure on it which is compatible with the metric. But what I am saying is that this does not work vice-versa.

Take the Riemann sphere defined by two charts $U_{\alpha }=U_{\beta }=\mathbb {C}$ and transition map $\varphi _{\alpha \beta }:z\mapsto 1/z$. How do you measure lengths there? We can certainly find a Riemannian metric compatible with this conformal structure, but not uniquely: We get a whole class of metrics, up to conformal equivalence. -- Momotaro (talk) 13:23, 6 May 2009 (UTC)

Oh, now I just remarked that conformal equivalence is even mentioned in the second definition, but only at the end. This makes it somewhat self-contradictory. -- Momotaro (talk) 13:26, 6 May 2009 (UTC)

I made an attempt to correct the second definition, should be ok now. -- Momotaro (talk) 13:10, 8 May 2009 (UTC)

I think it would be good to have a recipe in the article, e.g. how to get a Riemannian manifold from a Riemann surface. And which Riemannian metric is the preferred one. -- Spaetzle (talk) 15:22, 8 Apr 2011 (UTC)

There is no unique preferred one. As mentioned in the article, a Riemann surface has a unique Riemannian metric with constant curvature K ∈ {-1, 0, 1}; for many purposes this is the preferred one. But for many other purposes people prefer to view a compact Riemann surface as a projective curve given by a homogeneous polynomial in projective coordinates. This is sometimes possible in CP² and always possible in CP³. The Riemannian metric from such a representation is never of constant curvature, yet it makes the Riemann surface a minimal surface in its projective space. There are undoubtedly many other considerations.Daqu (talk) 01:19, 12 May 2012 (UTC)

Please, explain what the drawings of surface in the article represent

Do you think that it is clear,when you 'draw' the Riemann surface associated to the holomorphic function,for example,$f(z)={\sqrt {z}}$,what does this drawing represent ? A function drawn in a 3-dimension space must represent,in the simplest interpretation,a real function z in the two real variables x and y,so that z=f(x,y).A holomorphic function f in the complex variable z,should at least be represented graphically by TWO surfaces in a 3-dimension space,that is one for the real part and one for the imaginary part of f.

Don't you think that this elementary issue should be cleared up,before talking of anything about the Riemann surfaces ? —Preceding unsigned comment added by 90.0.205.206 (talk) 19:23, 22 March 2010 (UTC)

I gave it a shot. Better now? Hanche (talk) 22:56, 23 March 2010 (UTC)

Confusing statement should be removed

The fourth paragraph reads in full:

"Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties."

I love Riemann surfaces, but:

To claim that certain geometrical facts are "nice as possible" -- with or without the quotation marks around the word "nice" -- is just meaningless, and hence just confusing. If there is an underlying fact here, then why not just state it?

Certainly many Riemann surfaces have conformal automorphism groups that are much smaller than the largest possible symmetry group for a Riemannian surface of the same genus. This certainly isn't "as nice as possible", however the phrase is defined.

Nor is the inability to measure length, a staple of most forms of geometry.

Also, every compact Riemann surface is conformally equivalent to some (projective) algebraic curve, so it seems weird to compare them with "other" curves.Daqu (talk) 01:06, 12 May 2012 (UTC)

Self-intersection

As I understand it (undergraduate with limited analysis/topology experience), manifolds proper don't self-intersect or local homeomorphisms would fail, right (an X is not locally homeomorphic to a line, for instance)? So perhaps a note that sqrt(z) does not properly self-intersect, only its immersion in 3D, would be worthwhile beneath the page image. 67.79.154.194 (talk) 00:45, 15 December 2012 (UTC)

I think this highlights a serious problem with the diagram (and others in the article). The diagram illustrates a multi-valued function on the complex plane, and not a Riemann surface. This is likely to cause confusion for those that do not already understand what a Riemann surface is. A key point about a Riemann surface is that there is no vertical axis: it may be thought of as the domain of a function, not as the function itself. This misrepresentation implicit in the diagrams should really be addressed. — Quondum 06:22, 15 December 2012 (UTC)

What he said. 76.106.149.108 (talk) 19:12, 30 June 2013 (UTC)

Needed:

The Simple English Wikipedia article on this. 76.106.149.108 (talk) 19:11, 30 June 2013 (UTC)

Riemann Surface is not the 3D surface

There needs to be clarification in the article. The surfaces shown are either the real or imaginary parts. The whole surface is, in general, actually "4-dimensional" in the sense that it has 2 imaginary parts and 2 real parts. — Preceding unsigned comment added by 99.19.84.64 (talk) 16:57, 15 August 2013 (UTC)

I agree about clarification. What the depiction of, say, the square-root function sqrt(z) means is several things: The Riemann surface is a surface on which the function sqrt(z) can be defined in a single-valued manner. It can be thought of as having two points lying above each value of z on the complex plane (except only one point above z = 0). One of these points above z = r e^iθ for 0 <= θ < 2π represents a point z' at which sqrt(z') = sqrt(r) e^iθ/2, and the other point above this same z represents a point z at which sqrt(z) = sqrt(r) e^i(θ/2+π) (= -sqrt(r) e^i(θ/2+π)).

In this way, the Riemann surface depicts a new domain for a "new version" of the sqrt function, on which sqrt is single-valued instead of being multiple-valued (in this case, double-valued).

This is in contrast to having a "graph" like this where the vertical scale represents some concrete number with a particular meaning: in this kind of depiction of a Riemann surface, it does not.Daqu (talk) 21:19, 10 February 2015 (UTC)

I think the diagrams should be replaced as highly misleading. This has already been commented on above (see #Self-intersection). Daku's description is accurate; the Riemann surface shows both the real and imaginary parts of the domain, and the vertical axis should be removed. A Riemann surface is not a function. It is better thought of as a suitably glued-together paper surface lying completely flat (think of nearly collapsing the diagrams, but keeping the identity of the surfaces distinct). —Quondum 00:47, 11 February 2015 (UTC)

But it's not correct to claim (as in #Self-intersection) that a Riemann surface is not the depiction of a multi-valued function. Perhaps saying the Riemann surface of, say, sqrt(z) is the same as the depiction of that multi-valued function is slightly over-simplifying, but it is not wrong to say that. A very robust definition of the Riemann S surface of sqrt(z) is to define

S as {(z,sqrt(z)) in C²}, which is identical to saying S = {(u², u) in C²}.Daqu (talk) 23:51, 11 February 2015 (UTC)

I'm afraid this goes directly against my understanding of a Riemann surface, as explained by Roger Penrose. A Riemann surface is not a function; it is only the domain of a function. Your description earlier was good. A Riemann surface, by my understanding, can be considered as a pairing of a point in C with a discrete variable, which in this example would be from a two-element set. This would be like saying S = { (u, w) in C × {0,1} }. The Riemann surface that is needed as the domain of z^1/2 is the same Riemann surface that is needed as the domain of 5z^3/2. We can then define functions on S, e.g. sqrt : S → C. —Quondum 00:51, 12 February 2015 (UTC)

If you reread what I wrote, you will see that I defined a surface, not a function. The surface is of course related to a function, but I never said or implied that the Riemann surface is a function. As you say, it is not. Still, it can be used to depict one. The meaning of your comment about "pairing a function with a discrete variable" is not clear to me.

But also, defining a Riemann surface via a function is only one way to define it. Another slightly more modern way (though both are important!) is to say that a Riemann surface is a manifold whose transition functions are each a holomorphic function from one open set of C to another.Daqu (talk) 17:00, 18 February 2015 (UTC)

By "transition function" do you mean chart of an atlas? Probably not important; I think we are just somehow missing each other (for example, I cannot figure out what the set of pairs of complex numbers S is meant to represent in relation to a Riemann surface). Let's try another description: a Riemann surface is a covering space of the complex plane (with special treatment for isolated points). I think the important point is that the existing diagrams invite the reader into a misunderstanding that the vertical scale (and hence the shape and curvature) are part of the definition. —Quondum 01:01, 19 February 2015 (UTC)

Yes, he means charts. His pairs of complex numbers description is valid; it describes the Riemann surface embedded in a particular way in C². Constructing charts can be done using the implicit function theorem. Your description is as a branched cover of C. Constructing charts can be done by taking branch cuts. The two are related by the projection C² → C that sends (z, w) → z, because if z is known, then there are only two possible values for w, the two square roots of z.

I would also like to add that I think it's not a good idea to think of the branched cover description as pairs of points in C and a discrete variable. There is set-theoretically such a description, but the second variable ought to vary holomorphically in the first. The extrinsic description in C² has this property, for example. Ozob (talk) 03:20, 19 February 2015 (UTC)

I agree that the pairing with a discrete variable is unsuitable for use in the article; I was only trying to try to show that the description with two complex variables defines something with too much structure, though I'm clearly not being heard. Let's try another tack. The S = {(u², u) in C² description is a representation of a Riemann surface. A Riemann surface is not a graph (or function), whereas the typical reader looking at the diagram and captions would be inclined to assume that it is. —Quondum 17:20, 19 February 2015 (UTC)

I think you're using the word "representation" in an imprecise, non-mathematical sense. There is a completely precise way to capture the concept you want: The set S is a Riemann surface embedded in C². It is a complex submanifold. I'm not saying that we want to discuss general complex manifolds and their submanifolds in this article. All I'm saying is that S is a valid description of a Riemann surface. One can find charts on it, one can map it to C and view it as a branched cover, indeed, it's just as good as any other description. Ozob (talk) 02:50, 20 February 2015 (UTC)

I meant the term in a precise, mathematical sense. However, since this is clearly not showing signs of reaching any mutual understanding, I'll bow out of this discussion. —Quondum 03:47, 20 February 2015 (UTC)

As "deformed versions of the complex plane" ???

The article begins with the statement:

"Riemann surfaces can be thought of as deformed versions of the complex plane: . . .."

No, they can't be thought of this way — not without deeply confusing the thinker.Daqu (talk) 20:50, 10 February 2015 (UTC)

Dimension

The article states that "Riemann surface [...] is a one-dimensional complex manifold". But later it states that "Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface)". So how many dimensions does it have? Since it's a surface, it should be 2, right? 89.229.8.110 (talk) 20:45, 26 August 2017 (UTC)

(I know this question is 2 years old but...) A dimension d complex manifold is a dimension 2d real manifold (with a complex structure added). Complex manifold are locally modelled on C^d, and C^d is R^2d if we forget the complex structure. --Arnaud Chéritat (talk) 23:35, 26 December 2019 (UTC)

Connected or not

The definitions 1 and 2 are not fully equivalent: one mentions connectedness and not the other. --Arnaud Chéritat (talk) 23:37, 26 December 2019 (UTC)

Actually oriented manifolds are connected. Check the page on oriented manifolds. Julian Barathieu (talk) 17:08, 3 November 2021 (UTC)

Non-metrizable counterexample?

Paragraph three of the introduction states a necessary condition for a 2d manifold to be a Riemann surface - it must be orientable and metrizable. It then gives three counterexamples of non-orientable manifolds. Seems to me that the article would be improved by providing a non-metrizable counterexample. Not crucial to the article but an improvement.

Suggestions for non-metrizable counterexamples cheerfully accepted. Mr. Swordfish (talk) 23:45, 30 March 2021 (UTC)

metrisability for locally compact separable (in T3 sense) spaces is more or less equivalent to being second countable so the non-metrisable connected surfaces are exactly these. For a explicit example you can take the long line times a circle. jraimbau (talk) 12:13, 31 March 2021 (UTC)

Thanks. Long line cross S2 seems like an obvious counterexample (obvious once you point it out, that is). Should we mention it in the article? We'd need a cite, of course. Mr. Swordfish (talk) 23:38, 31 March 2021 (UTC)

Lecture notes by Eynard

I have tried to add the following lecture notes from Arxiv to the References:

• Eynard, Bertrand (2018), "Lectures notes on compact Riemann surfaces", arXiv.org, retrieved 2021-11-04

This was reverted by User:D.Lazard due to WP:reliable sources concerns. However, these are lecture notes by a known expert, author of this book. Isn't this OK per Wikipedia:Reliable_sources#Self-published_sources_(online_and_paper)? Sylvain Ribault (talk) 13:15, 4 November 2021 (UTC)

This text has been submitted in 2018 to a physical journal, and is still not published. I do not know the reason for this non-publication, but the name of the author is not sufficient for verifying whether this text is worth to be cited (not all papers of Field medals are good articles).

Also, this citation has been added in a section that contains 9 published books, including 5 that have "Riemann surface" in their title. This seems sufficient for sourcing the article. So, adding a new unpublished item would make sense only if it contains some material that is not in the other sources. This is not the case, as the citation is not presented as containing new results.

In summary, for adding this citation, strong reasons must be explicited, which are currently lacking. D.Lazard (talk) 14:04, 18 November 2021 (UTC)

Thank you for the elaboration. I would have liked to include some openly accessible general reference. I will now leave it as a suggestion. Sylvain Ribault (talk) 20:36, 19 November 2021 (UTC)

in the first image, after rotation colour = height, really?

The description of the first image says "For this function, it is also the height after rotating the plot 180° around the vertical axis." ("it" refers to the color).

I don't see how a rotation in the vertical axis would achieve this. Atoll (talk) 07:31, 2 March 2023 (UTC)

The big picture at the top.

It's described as, "An open subset of the Riemann surface for the multivalued function..."

But it's not on open subset. It's a bounded, closed subset. Every edge is a sharp edge.

Should I fix it or am I wrong? I'm not confident I can tell.

Verdana♥Bold 17:11, 18 June 2024 (UTC)

It's an open set, because a Riemann surface is two dimensional. (I.e., it's not open in R^3, but is open in the surface.) Tito Omburo (talk) 22:05, 18 June 2024 (UTC)

Since it is not said whether the represented surface contains its boundary, one cannot say whether the represented surface is open or closed in the Riemann surface, and this has the same low importance as discussing whether the plot of a real function contains the points where the curve is truncated. I have removed "open subset" from the caption, and added an explanation of the meaning of the coordinates. D.Lazard (talk) 01:36, 19 June 2024 (UTC)

Top image description

@Tito Omburo: no, the real part is not represented by colour in the top image. Just look at the image: the colour is constant along the radial coordinate but changes along the vertical axis. It represents the imaginary part, too. If it represented the real part, as the current description erroneously says, it would be the other way round: constant along the vertical axes, non-constant along the radial coordinate. colt_browning (talk) 17:51, 2 September 2024 (UTC)

My mistake, I see that the color represents argument in that image. I have replaced it with a more appropriate image, with color given by the real part. Tito Omburo (talk) 11:27, 3 September 2024 (UTC)