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Take a look at the short paragraph below:
(Another reader notes that afaik this article has the stereographic projection wrong - the sphere and the plane should not have the same origin. The sphere rests on the plane and the sphere point is defined as the 'other' intersection of the line with the sphere. 62.49.9.94
The complex manifold structure on the Riemann sphere is specified by an atlas with two charts and coordinates z and w
The transition function between the two patches is w = 1/z, which is clearly holomorphic and so defines a complex structure. To see that these charts give the topology of the sphere note that we can give an atlas on S2 by stereographic projection onto the complex planes tangent to the north and south poles respectively. Labeling points in S2 by (x1, x2, x3) where , we have
which satisfies the equation w = 1/z. In terms of standard spherical coordinates (θ, φ)
are not in the proper context, they implicitely assume that the connection between the Riemann sphere and the normal sphere is already established, while it was not.
Also, in a paragraph below in text one says:
Same problem. This makes no sense until a correspondence between the normal sphere and the Riemann sphere is explicitely stated, and that correspondence is fixed, so there is no ambiguity. Oleg Alexandrov 01:57, 15 Feb 2005 (UTC)
Perhaps the wording in the article is not the best. If you think things would be clearer, feel free to make it more rigorous. The idea is very simple though. You have two coordinate charts (both copies of C) with coordinates z and w. The transition function on the overlap (where both z and w are nonzero) is given by z = 1/w. That's all the information needed to specify the complex manifold. The equations relating z and w to the x's is defining the relationship between the geometrical unit 2-sphere and the Riemann sphere. -- Fropuff 04:23, 2005 Feb 15 (UTC)
No worries. It is a little confusing with C serving as both part of the manifold and as coordinate charts. One can be clever with the notation but it is perhaps more confusing in the end. -- Fropuff 02:30, 2005 Feb 16 (UTC)
That is what I meant. But as you say, there is little that can be done about it. I think what is there now works fine. -- Fropuff 03:01, 2005 Feb 16 (UTC)
(No puns intended on complexity or angle)
I have found several much simpler explanations for the complex sphere. Examples include the one in Penrose's The Road to Reality or even the more technical one in Needham's Visual Complex Analysis. Their common simplifying factor is presenting it in terms of geometry, with diagrams to boot.
Presenting the Riemann sphere in terms of a set theory and mappings seems unnecessarily abstract. Giving the transformations and set-theoretic definitions, IMO, should be secondary to giving the geometric derivations of mapping the complex plane (+ infinity) to points on a sphere.
Disclaimer: I am a high school senior learning these things on my own. I don't know how textbooks do it. I just look at what I'm given and judge. In the case of Penrose and Needham, I say "this makes sense and is really cool!" But for Wikipedia, I come out of it more like "huh? Why do it this way? What the heck does this even have to do with spheres?"
So basically, I don't pretend to be an authority on this. —Preceding unsigned comment added by DomenicDenicola ( talk • contribs)
(Notes: this is meant to be standalone, so when it's done we would find overlap from other sections and remove it here or there.)
Define (i.e. the complex numbers joined with the point at infinity). The Riemann sphere is based on the transformation from to and is in the form
where and .
We visualize the Riemman sphere as a sphere in 3-space, i.e. in . Every point on the sphere has both a value and value, related by the above transformation. That is, transforms the sphere onto itself.
To establish the correspondence between points in the extended complex plane and the Riemann sphere, we first place the plane across the sphere's equator. We then use stereographic projection from the south pole of the sphere. This is done by drawing a line from the south pole that intersects both the sphere and the complex plane; a unique, one-to-one correspondence is then established between points on the complex plane and points on the Riemann sphere. Note that points on the complex plane inside the unit circle will map to the upper hemisphere, and points outside will map to the lower hemisphere.
In order to complete this one-to-one correspondence for the extended complex plane, we define the south pole to be . Note that the north pole is .
The correspondence between the plane and the Riemann sphere is done in much the same way, simply "upside down." That is, the plane is an equitorial plane oriented oppositely to the plane, such that matches to . We then perform the stereographic projection from the north pole, and similarly define the north pole to be . Now, every point on the sphere has both a and coordinate, related by the transformation above.
The equator of the sphere is the unit circle in the complex plane; in a similar fashion, circles can be found for the imaginary line and real lines. Note that these are shared between the two projections, because the relation is holomorphic.
This is a specific case of how stereographic projection maps all lines and circles in the complex plane to circles on the Riemann sphere. The reason that lines are mapped to circles is that a line with infinite length can simply be thought of as a circle that passes through the point at infinity.
Möbius transformations, which send to , are often visualized as acting on the Riemann sphere. They are in the form
where , , and . They map the sphere to itself, preserving important features such as angles and circles/lines. This is because they are only composed of dilations, translations and inversions.
I think I've confused myself with the whole Möbius transformation thing. I believe my explanation is only valid for the transformation . So how do Möbius transformations fit in? I know they map Riemann spheres to Riemann spheres, but I don't think they do so in the manner I described (simple projection through the matched planes).
Does this sentence even make sense? I might not be saying what I'm trying to say... "Note that these are shared between the two projections, because the Möbius transformation is holomorphic."
Comments on explanation? I really like it, but hey, that's because it makes sense to me.
I really need diagrams, especially ones just showing the sphere superimposed with at least the plane and with labelled.
Domenic Denicola 19:50, 9 December 2005 (UTC)
-- Fropuff 21:15, 9 December 2005 (UTC)
I deleted these sentences because I wasn't sure they worked for tangent planes:
If they only fit in with the equitorial projection, I could add them in the section on that; alternatively, if they work in both, I'll restore them to the appropriate places. I need guidance, however.
The planes are still switched in orientation, right?
Hope everyone likes the results!
Domenic Denicola 06:32, 13 December 2005 (UTC)
This article leaves me utterly confused. I don't expect to grasp the formulas toward the bottom of the page in any math-related article; I don't have sufficient background. But none of this makes any sense at all, not from the first few words.
How is a Riemann sphere distinct from an ordinary sphere? Explain it so my daughter can understand it. She knows the difference between a sphere and a circle; is this the same thing? Circle, sphere, Riemann sphere? (Square, cube, hypercube.)
Is there any difference between Riemann sphere and Riemann space? If so, what? Say it without any numbers or special words. If I lived on the surface of a Riemann sphere, what would be different about my life? Would all my doughnuts turn into coffeecups? Are squares still square?
Is there more than one Riemann sphere? Can they come in different sizes? (I suspect the answers are no and no.) Could I even tell if I did live in a Riemann sphere? It looks to me, offhand, as if the thing is of infinite size. Wouldn't any finite zone or section (my local known universe) always appear Euclidean? Is any point on the Riemann sphere distinguished? If so, what would happen if I stood there? Would I blow up? Would my left and right hands disappear or get stuck together? Could I even tell?
What is the use of this thing? How can I apply it and to what? Automotive engineering? Faster-than-light spacecraft theory? Molecular biology? Game theory? Can I win a bar bet with it; if so, what's the bet?
I came here because graphics were requested; I'm willing to do graphics. But first, somebody will have to tell me what it is. John Reid 20:14, 14 April 2006 (UTC)
The fine points may be lost on a layman but please have the courtesy not to look down your nose at me. I have enough background in topology to understand why I can remove my vest without taking off my coat. I know that if I glue together all four corners of a checkerboard and glue (along each edge) four squares to four squares then I have created a bag-like thing which is not equivalent to a sphere; it is not even a manifold because there are distinguished points. I know why I can never comb a hairy ball. And I even have a pretty fair idea why the N-color problem has a different solution on a torus. If you can't explain the subject of this article to me, I think you don't have a clear grasp of the concept -- only what you have been told. No offense intended.
The question about different sizes makes a great deal of sense. One of the first things I do when trying to understand a new concept is to search for limits. Is there one only or more than one? Does it come in more than one size or variety? As I said, I suspect there is only one, of infinite size. You still have not told me if the point at infinity is distinguished. Indeed, you've failed to answer most of my questions. That's okay; but if you don't know the answers then please don't be so dismissive of the questions.
Nothing useful is purely abstract. Experts who work with a concept regularly may be comfortable with an abstract representation; but there is always some connection to reality -- otherwise the concept is nonsensical. You may have difficulty associating this topic to a practical aspect of life but I suggest that there is such a connection. If not, I'm tempted to say that however important it may be to a specialist, it holds no possible interest for the general reader. John Reid 22:20, 14 April 2006 (UTC)
To reply to John Reid, from the top down, here it goes. The Riemann sphere is just the complex plane with an extra point added in, called the point at infinity. For analogy, look at the real line. There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit.
In the same way, one can pretend that all rays in the complex plane originating from 0 actually have an endpoint, and they all eventually meet at infinity, a point far-far away (not accurate as Elroch mentions above, but helpful in imagining things).
The Riemann sphere is not the same as the usual sphere, but they are topologically equivalent. Imagine a normal sphere, remove the north pole, and make the obtained hole there larger and larger (assume the sphere is made of very flexible rubber). Eventually, that sphere without a point can be flattened in a plane, the complex plane. The original north pole corresponds to the point at infinity in the complex plane.
There is only one Riemann sphere, as the point at infinity is just a symbol, its actual nature is not relevant. In the same way that there exists essentially one normal sphere. The radiuses may differ, but any sphere can be deformed gently into another sphere, without tearing the surface. In exactly the same way a sphere is the same as the surface of a cube, but not with the surfce of a donut.
You can't say if any portion of the Riemann sphere appears Euclidean, or whether it is infinite in size or not. That because there is no concept of distance and size on the Riemann sphere. Any portion of the sphere can be stretched/shrank in any way as long as the sphere does not burst or separate patches merge.
The Riemann sphere does not get applied directly much beyond math, or otherwise I never heard of it. It is a useful construct, but rather abstract.
Above I talked about the topology of the Riemann sphere, not its differential geometry . But that would be harder to explain.
I don't know how satisfactory you found the answers. Try to read them though, and let me know if you have questions. Oleg Alexandrov ( talk) 02:37, 16 April 2006 (UTC)
Elroch made a lot of good edits to this article. It is now more mathematically correct, but it is hard to understand for somebody not knowing math however. I believe the geometric viewpoint, which, if not entirely accurate, was helpful in illustrating what is going on. Wonder what you think. Oleg Alexandrov ( talk) 15:09, 17 April 2006 (UTC)
This article (like the Stereographic projection article) prefers the projection onto a plane tangent at a pole over the projection onto a plane through the equator. It even asserts that the former is more popular. This contradicts my experience. Among the texts in front of me, the equatorial version is used books by Rudin, Bredon, Thorpe, Oprea, and Brown/Churchill; the pole-tangent version is used in Do Carmo; Spivak uses both versions.
More importantly, this article contradicts itself. Its claims about the transition maps being , corresponding to , etc. are all assuming the equatorial version. (In the equatorial version, the unit circle in is sent to the equator; in the pole-tangent version, the circle of radius 2 is sent there, so the unit circles from the two complex planes don't match up.)
In short, while the pole-tangent version of stereographic projection has uses in differential geometry, the equatorial version seems unequivocally better suited to complex analysis, which is where this article comes in. So unless there are objections I'm going to start making these changes. Joshua R. Davis 19:55, 25 April 2007 (UTC)
I appreciate that mathematical experts may be writing this article from their own minds, without referring to source material. I've done the same in articles where I have expertise. This is Wikipedia though, and at least a few citations are needed. At a minimum, cite a few books or papers that describe a Riemann sphere. Davidwr 15:10, 3 May 2007 (UTC)
Hello all. I just added applications and references to the article. But in fact I have rewritten the entire article. I think the new version is more precise/rigorous, clearer about what the metric is and what depends on the metric, better organized, less redundant, etc. (But then I would think that. :) I have already transplanted some of the purely geometric material to Stereographic projection (also newly rewritten).
Because there has been recent activity on this article and this talk page (moreover by editors whom I know and respect) I don't want to impose a complete rewrite on everyone out of the blue, so I'm posting it here for comments/incorporation. Joshua R. Davis 04:18, 4 May 2007 (UTC)
In mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as
well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted .
On a purely algebraic level, the complex numbers with an extra infinity element constitute the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one- dimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.
As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane . Let and be complex coordinates on . Identify the nonzero complex numbers with the nonzero complex numbers using the transition maps
Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere.
Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane.
In other words, (almost) every point in the Riemann sphere has both a value and a value, and the two values are related by . The point where should then have -value ""; in this sense, the origin of the -chart plays the role of "" in the -chart. Symmetrically, the origin of the -chart plays the role of with respect to the -chart.
Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified with . On the other hand, the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
The uniformization theorem, a central result in the classification of Riemann surfaces, states that the only simply-connected one-dimensional complex manifolds are the complex plane, the hyperbolic plane, and the Riemann sphere. Of these, the Riemann sphere is the only one that is closed ( compact and boundaryless).
The Riemann sphere can also be defined as the complex projective line. This is the subset of consisting of all pairs of complex numbers, not both zero, modulo the equivalence relation
for all nonzero complex numbers . The complex plane , with coordinate , can be mapped into the complex projective line by
Another copy of with coordinate can be mapped in by
These two complex charts cover the projective line. For nonzero the identifications
demonstrate that the transition maps are and , as above.
This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line or smooth conic in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere's automorphisms, later in this article.
The Riemann sphere can be visualized as the unit sphere in the three-dimensional real space . To this end, consider the stereographic projection from the unit sphere minus the point onto the plane , which we identify with the complex plane by . In Cartesian coordinates and spherical coordinates on the sphere (with the zenith and the azimuth), the projection is
Similarly, stereographic projection from onto the plane, identified with another copy of the complex plane by , is written
(The two complex planes are identified differently with the plane . An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.) The transition maps between -coordinates and -coordinates are obtained by composing one projection with the inverse of the other. They turn out to be and , as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere.
Under this diffeomorphism, the unit circle in the -chart, the unit circle in the -chart, and the equator of the unit sphere are all identified. The unit disk is identified with the southern hemisphere , while the unit disk is identified with the northern hemisphere .
A Riemann surface does not come equipped with any particular Riemannian metric. However, the complex structure of the Riemann surface does uniquely determine a metric up to conformal equivalence. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positive smooth function.) Conversely, any metric on an oriented surface uniquely determines a complex structure, which depends on the metric only up to conformal equivalence. Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface.
Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with constant curvature in any given conformal class.
In the case of the Riemann sphere, the Gauss-Bonnet theorem implies that a constant-curvature metric must have positive curvature K. It follows that the metric must be isometric to the sphere of radius in via stereographic projection.
In the -chart on the Riemann sphere, the metric with is given by
In real coordinates , the formula is
Up to a constant factor, this metric agrees with the standard Fubini-Study metric on complex projective space (of which the Riemann sphere is an example).
Conversely, let S denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on S. It follows that any metric on S is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since "roundness" is not an invariant of conformal geometry. The Riemann sphere is only a conformal manifold not a Riemannian manifold. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice.
The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form
where , , , and are complex numbers such that . Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.
The Möbius transformations are profitably viewed as transformations on the complex projective line. In projective coordinates, the transformation is written
Thus the Möbius transformations correspond to complex matrices with nonzero determinant. These are the projective linear transformations .
If one endows the Riemann sphere with the Fubini-Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of , namely . It is isomorphic to the rotation group , which is the isometry group of the unit sphere in .
Overall, I think this is a big improvement. It's very nicely organized and well written. My main criticism is with the section "As a sphere". This section should really start out with a description of the stereographic projection rather than its inverse. The complex coordinates should be given by
(which are incidentally much simpler and easier to grok than their inverses). This shows very geometrically how to put complex coordinates onto the sphere to make it into a complex manifold. One can then give the inverse transformations to complete the picture.
There is a subtlety that occurs in these equations which should be explained. When projecting from opposite poles, one is not quite projecting onto the same plane. The planes are complex conjugates of each other. This is necessary to make the transition functions holomorphic rather than antiholomorphic. I remember this confused me a good deal when I first encountered the Riemann sphere. I think it is also the reason I, at one time, preferred projections onto the polar tangent planes: as the planes were distinct it was less easy to confuse them.
Minor quips: I prefer using z and w for the coordinates rather than ξ and ζ (which, by the way, appear nearly identical with my fonts), but I can see how this might lead to confusion. Also, for reasons unbeknownst to me, it seems much more natural to complex conjugate the southern hemisphere chart (i.e. when projecting from the north pole) rather than the northern one so that
Maybe its because I live in the northern hemisphere and I'm biased. But again this is a personal preference and I'm not bound to it. -- Fropuff 08:11, 4 May 2007 (UTC)
Another thought regarding the metric: the way you've written it makes it sound like the round metric is chosen arbitrarily, which is, of course, far from the truth. Firstly, the metric is the pullback of the round metric on the sphere by the inverse stereographic projection. Secondly, the complex structure on the sphere uniquely determines a conformal class of metrics on the sphere. Within this class there is a unique (or so I recall) metric with constant curvature. Normalize the curvature to be +1 and you get the round metric. Perhaps it would be best to put all the stuff about the metric in its own section. -- Fropuff 08:43, 4 May 2007 (UTC)
If we want to stick with Greek letters for complex variables then ζ and ξ are fine, and like it or not, your choice of which plane to conjugate does seem more common in the literature so I guess I'll have to live with it. The only thing I really miss from the present article is a description of the stereographic projection in spherical coordinates. In present notation we should have
where θ is the zenith angle and φ is the azimuth angle. I find these formulas extremely useful. They show clearly that circles of latitude on the sphere map to circles on the plane while lines of longitude map to radial lines. I realize these formulas are at stereographic projection but its nice to see them in complex notation. Other than that I am happy to replace the article with the new version. -- Fropuff 00:18, 7 May 2007 (UTC)
Why do we have this image? What does it half to do with the Riemann sphere? It's relevence is never explained. As far as I can tell, only the title of the image has anything to do with the article. The Riemann sphere is topologically a sphere, but has no one intrinsic geometry (you can get a metric by pulling back a metric with stereographic projection, but even then there is no unique choice of projection). Loxodromes are intrinsically geometric objects, and so would seem to have little to do with the Riemann sphere. And even if you do give the Riemann sphere a metric, the obvious choice would be a pullback via stereographic projection, and even then a loxodrome wouldn't come from any natural curve on the complex plane. The curve that maps to a loxodrome would be a "spiral" starting out at the origin and winding around finitely many times before shooting out to infinity (as opposed to natural spirals, which wrap around infinitely often). No natural function does that. So the image strikes me as a complete nonsequiter, and even if it isn't, it needs to be tied into the article somehow. skeptical scientist ( talk) 12:41, 1 July 2007 (UTC)
I cannot deduce from the formulaes in the article how to transform a point P=x+i·y of the Gaussian plane onto the x,y,z Cartesian coordinates (or longitude,latitude spherical coordinateS) of the Riemann sphere. :-/ Is it that complicated? -- RokerHRO ( talk) 09:50, 18 May 2010 (UTC)
∞ + ∞ = ∞ is stated at the section Arithmetic operations. I think this operation must be moved to the undefined operations (together with ∞ - ∞ and 0·∞). You can find a counterexample at page 30 here. I'm not an expert, so I'm afraid I may not be correct and I didn't change it. So please, if someone agrees, change it.-- Ssola ( talk) 17:57, 7 March 2012 (UTC)
This article is to dispell the obvious mistake made in the Riemann sphere definition which states that "∞" infinity is near to very large numbers and that zero "0" is near to very small numbers. This is false for two reasons: 1) nothing and infinity are unlimited concepts and are therefore by their very nature completely incompatible mathematically and conceptually with limited numbers. 2) The concept that zero is close to small numbers and infinity is close to large numbers is based on the assumants inability to conceive of something infinite outside of space and time resulting in the assumants mind counting up or down forever caught in a limited process that given enough time would never reach its infinite goal for the simple fact that it is always limited.
Conclusion this theory is logically refuted as false. — Preceding unsigned comment added by 193.200.145.253 ( talk) 10:52, 23 July 2013 (UTC)
The article addresses the problem of division by zero and rules out the special cases ∞ + ∞, ∞ - ∞ and 0 ⋅ ∞ as undefined. But what about 0/0? Is it still left undefined, as it is with the real numbers? My intuition tells me it would... SBareSSomErMig ( talk) 07:04, 25 September 2013 (UTC)
Why does this article say infinity plus infinity is undefined?? It has always made sense to define it as infinity. Infinity plus negative infinity is undefined though. Georgia guy ( talk) 22:00, 16 January 2014 (UTC)
Some parts of this article seem to imply any stereographic projection of the complex plane to a sphere is "a Riemann sphere", whereas other parts seem to imply there is only one canonical projection which forms "the Riemann sphere". Some consistency would be nice here.
My intuition from elsewhere in mathematics is that the former is true, even though the wording of the latter is more likely (e.g. in computational complexity we may refer to "the first Turing machine enumerated in shortlex form as a member of {0, 1}*", ignoring the fact that which Turing machine this may be depends entirely upon the coding language being used to model the machine in its coded form--generally because it has been proven that the choice of coding method is immaterial as long as all possible machines can be enumerated). TricksterWolf ( talk) 03:31, 6 February 2014 (UTC)
Any two stereographic projections give the same complex structure on the sphere (and also the same conformal structure). So if one thinks of the Riemann sphere as a complex or conformal manifold only, then there is no preferred projection. However, if one thinks of the Riemann sphere as the extended complex plane, then there is a preferred point at infinity and a preferred stereographic projection. Which of these is intended depends on the context. (Ironically, one of the participants in this very discussion has a username that illustrates both of these cases, with g=0 and different values of n.) Sławomir Biały ( talk) 16:33, 6 August 2014 (UTC)
It is claimed in the Rational functions section that "the set of complex rational functions ... form all possible holomorphic functions from the Riemann sphere to itself". As far as I know, is not a rational function, yet it is holomorphic. So this claim appears to be false. The only way out I can see is if we consider to be a "polynomial" on the grounds that it has a power series, but this seems to be a stretch. Am I mistaken? Luqui ( talk) 05:19, 8 July 2018 (UTC)
It seems to be that the topic of the extended complex plane is a bit primary to the concept of the “Riemann sphere” per se, and subordinating the former to the latter seems a bit backward.
We might start by defining the extended complex plane as (a) the complex numbers plus an extra point at infinity with defined topologically to be the center of balls , but it could alternately be defined as (b) equivalence classes of pairs of complex numbers such that iff i.e. the ratios or (c) some atlas of charts, e.g. two separate stereographic projections of the sphere from opposite poles onto the complex plane.
Then a section about the “Riemann sphere” (which should perhaps better be named after Carl Neumann, Riemann's student who came up with / promoted the idea) could discuss using chord length or arclength (from inverse stereographic projection) as a distance function between arbitrary points in the extended complex plane, and could discuss the use of complex numbers as a sometimes convenient representation for studying spherical geometry.
But the notion of the extended complex numbers doesn't seem to inherently require thinking about a "sphere" or spherical distance. The Möbius transformations of the extended complex plane are a broader group than only the isometries (rotations) of the sphere, and also include translations of the plane ("parametric transformations"), scaling ("hyperbolic transformations")
– jacobolus (t) 23:33, 24 January 2023 (UTC)
On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states that every simply-connected Riemann surface is biholomorphic to the complex plane, the hyperbolic plane, or the Riemann sphere.It seems to me like any article about the extended complex numbers should be talking extensively about how as a number system they are commonly used to model points the 2-sphere, Euclidean plane, and hyperbolic plane (for better or worse; arguably there’s at least a bit of conceptual mismatch). But in an article entitled “Riemann sphere” those don’t quite seem to belong. – jacobolus (t) 06:05, 25 January 2023 (UTC)
The section As a sphere contains this paragraph:
"In order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point and the second except the point . Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at . Note that the two complex planes are identified differently with the plane . An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic."
This paragraph cries out to be rewritten clearly and accurately.
Already its first sentence describes stereographic projection exactly backwards.
Nobody has any idea what "intuitively seen as glued back-to-back" means.
In the phrase "... the two complex planes are identified differently with the plane", it is entirely unclear what "the plane" refers to.
And in the comment about "orientation reversal", there is no mention of what it is that has its orientation reversed, or where complex conjugation enters in.
(Is there really anything that gets its orientation reversed? Seems extremely doubtful.)
One of the worst paragraphs I've seen in a Wikipedia mathematics article.
— Preceding unsigned comment added by 2601:200:c082:2ea0:7185:a86f:eb51:745a ( talk) 11:38, 29 May 2023 (UTC)
To split or not to split? The page deals with a particular one-dimensional complex manifold known either by the stereographic projection of the page title or as the complex projective line. Every simply connected, one-dimensional complex manifold is isomorphic to this one, or the included plane, or the disk. On the side of sphere there is the realm of Riemann surfaces as a superset, whereas on the side of the projective line there is projective geometry. The combination of these two branches in one article corresponds an expectation that readers see that the two branches stem from the same trunk. — Rgdboer ( talk) 01:43, 28 October 2023 (UTC)
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Take a look at the short paragraph below:
(Another reader notes that afaik this article has the stereographic projection wrong - the sphere and the plane should not have the same origin. The sphere rests on the plane and the sphere point is defined as the 'other' intersection of the line with the sphere. 62.49.9.94
The complex manifold structure on the Riemann sphere is specified by an atlas with two charts and coordinates z and w
The transition function between the two patches is w = 1/z, which is clearly holomorphic and so defines a complex structure. To see that these charts give the topology of the sphere note that we can give an atlas on S2 by stereographic projection onto the complex planes tangent to the north and south poles respectively. Labeling points in S2 by (x1, x2, x3) where , we have
which satisfies the equation w = 1/z. In terms of standard spherical coordinates (θ, φ)
are not in the proper context, they implicitely assume that the connection between the Riemann sphere and the normal sphere is already established, while it was not.
Also, in a paragraph below in text one says:
Same problem. This makes no sense until a correspondence between the normal sphere and the Riemann sphere is explicitely stated, and that correspondence is fixed, so there is no ambiguity. Oleg Alexandrov 01:57, 15 Feb 2005 (UTC)
Perhaps the wording in the article is not the best. If you think things would be clearer, feel free to make it more rigorous. The idea is very simple though. You have two coordinate charts (both copies of C) with coordinates z and w. The transition function on the overlap (where both z and w are nonzero) is given by z = 1/w. That's all the information needed to specify the complex manifold. The equations relating z and w to the x's is defining the relationship between the geometrical unit 2-sphere and the Riemann sphere. -- Fropuff 04:23, 2005 Feb 15 (UTC)
No worries. It is a little confusing with C serving as both part of the manifold and as coordinate charts. One can be clever with the notation but it is perhaps more confusing in the end. -- Fropuff 02:30, 2005 Feb 16 (UTC)
That is what I meant. But as you say, there is little that can be done about it. I think what is there now works fine. -- Fropuff 03:01, 2005 Feb 16 (UTC)
(No puns intended on complexity or angle)
I have found several much simpler explanations for the complex sphere. Examples include the one in Penrose's The Road to Reality or even the more technical one in Needham's Visual Complex Analysis. Their common simplifying factor is presenting it in terms of geometry, with diagrams to boot.
Presenting the Riemann sphere in terms of a set theory and mappings seems unnecessarily abstract. Giving the transformations and set-theoretic definitions, IMO, should be secondary to giving the geometric derivations of mapping the complex plane (+ infinity) to points on a sphere.
Disclaimer: I am a high school senior learning these things on my own. I don't know how textbooks do it. I just look at what I'm given and judge. In the case of Penrose and Needham, I say "this makes sense and is really cool!" But for Wikipedia, I come out of it more like "huh? Why do it this way? What the heck does this even have to do with spheres?"
So basically, I don't pretend to be an authority on this. —Preceding unsigned comment added by DomenicDenicola ( talk • contribs)
(Notes: this is meant to be standalone, so when it's done we would find overlap from other sections and remove it here or there.)
Define (i.e. the complex numbers joined with the point at infinity). The Riemann sphere is based on the transformation from to and is in the form
where and .
We visualize the Riemman sphere as a sphere in 3-space, i.e. in . Every point on the sphere has both a value and value, related by the above transformation. That is, transforms the sphere onto itself.
To establish the correspondence between points in the extended complex plane and the Riemann sphere, we first place the plane across the sphere's equator. We then use stereographic projection from the south pole of the sphere. This is done by drawing a line from the south pole that intersects both the sphere and the complex plane; a unique, one-to-one correspondence is then established between points on the complex plane and points on the Riemann sphere. Note that points on the complex plane inside the unit circle will map to the upper hemisphere, and points outside will map to the lower hemisphere.
In order to complete this one-to-one correspondence for the extended complex plane, we define the south pole to be . Note that the north pole is .
The correspondence between the plane and the Riemann sphere is done in much the same way, simply "upside down." That is, the plane is an equitorial plane oriented oppositely to the plane, such that matches to . We then perform the stereographic projection from the north pole, and similarly define the north pole to be . Now, every point on the sphere has both a and coordinate, related by the transformation above.
The equator of the sphere is the unit circle in the complex plane; in a similar fashion, circles can be found for the imaginary line and real lines. Note that these are shared between the two projections, because the relation is holomorphic.
This is a specific case of how stereographic projection maps all lines and circles in the complex plane to circles on the Riemann sphere. The reason that lines are mapped to circles is that a line with infinite length can simply be thought of as a circle that passes through the point at infinity.
Möbius transformations, which send to , are often visualized as acting on the Riemann sphere. They are in the form
where , , and . They map the sphere to itself, preserving important features such as angles and circles/lines. This is because they are only composed of dilations, translations and inversions.
I think I've confused myself with the whole Möbius transformation thing. I believe my explanation is only valid for the transformation . So how do Möbius transformations fit in? I know they map Riemann spheres to Riemann spheres, but I don't think they do so in the manner I described (simple projection through the matched planes).
Does this sentence even make sense? I might not be saying what I'm trying to say... "Note that these are shared between the two projections, because the Möbius transformation is holomorphic."
Comments on explanation? I really like it, but hey, that's because it makes sense to me.
I really need diagrams, especially ones just showing the sphere superimposed with at least the plane and with labelled.
Domenic Denicola 19:50, 9 December 2005 (UTC)
-- Fropuff 21:15, 9 December 2005 (UTC)
I deleted these sentences because I wasn't sure they worked for tangent planes:
If they only fit in with the equitorial projection, I could add them in the section on that; alternatively, if they work in both, I'll restore them to the appropriate places. I need guidance, however.
The planes are still switched in orientation, right?
Hope everyone likes the results!
Domenic Denicola 06:32, 13 December 2005 (UTC)
This article leaves me utterly confused. I don't expect to grasp the formulas toward the bottom of the page in any math-related article; I don't have sufficient background. But none of this makes any sense at all, not from the first few words.
How is a Riemann sphere distinct from an ordinary sphere? Explain it so my daughter can understand it. She knows the difference between a sphere and a circle; is this the same thing? Circle, sphere, Riemann sphere? (Square, cube, hypercube.)
Is there any difference between Riemann sphere and Riemann space? If so, what? Say it without any numbers or special words. If I lived on the surface of a Riemann sphere, what would be different about my life? Would all my doughnuts turn into coffeecups? Are squares still square?
Is there more than one Riemann sphere? Can they come in different sizes? (I suspect the answers are no and no.) Could I even tell if I did live in a Riemann sphere? It looks to me, offhand, as if the thing is of infinite size. Wouldn't any finite zone or section (my local known universe) always appear Euclidean? Is any point on the Riemann sphere distinguished? If so, what would happen if I stood there? Would I blow up? Would my left and right hands disappear or get stuck together? Could I even tell?
What is the use of this thing? How can I apply it and to what? Automotive engineering? Faster-than-light spacecraft theory? Molecular biology? Game theory? Can I win a bar bet with it; if so, what's the bet?
I came here because graphics were requested; I'm willing to do graphics. But first, somebody will have to tell me what it is. John Reid 20:14, 14 April 2006 (UTC)
The fine points may be lost on a layman but please have the courtesy not to look down your nose at me. I have enough background in topology to understand why I can remove my vest without taking off my coat. I know that if I glue together all four corners of a checkerboard and glue (along each edge) four squares to four squares then I have created a bag-like thing which is not equivalent to a sphere; it is not even a manifold because there are distinguished points. I know why I can never comb a hairy ball. And I even have a pretty fair idea why the N-color problem has a different solution on a torus. If you can't explain the subject of this article to me, I think you don't have a clear grasp of the concept -- only what you have been told. No offense intended.
The question about different sizes makes a great deal of sense. One of the first things I do when trying to understand a new concept is to search for limits. Is there one only or more than one? Does it come in more than one size or variety? As I said, I suspect there is only one, of infinite size. You still have not told me if the point at infinity is distinguished. Indeed, you've failed to answer most of my questions. That's okay; but if you don't know the answers then please don't be so dismissive of the questions.
Nothing useful is purely abstract. Experts who work with a concept regularly may be comfortable with an abstract representation; but there is always some connection to reality -- otherwise the concept is nonsensical. You may have difficulty associating this topic to a practical aspect of life but I suggest that there is such a connection. If not, I'm tempted to say that however important it may be to a specialist, it holds no possible interest for the general reader. John Reid 22:20, 14 April 2006 (UTC)
To reply to John Reid, from the top down, here it goes. The Riemann sphere is just the complex plane with an extra point added in, called the point at infinity. For analogy, look at the real line. There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit.
In the same way, one can pretend that all rays in the complex plane originating from 0 actually have an endpoint, and they all eventually meet at infinity, a point far-far away (not accurate as Elroch mentions above, but helpful in imagining things).
The Riemann sphere is not the same as the usual sphere, but they are topologically equivalent. Imagine a normal sphere, remove the north pole, and make the obtained hole there larger and larger (assume the sphere is made of very flexible rubber). Eventually, that sphere without a point can be flattened in a plane, the complex plane. The original north pole corresponds to the point at infinity in the complex plane.
There is only one Riemann sphere, as the point at infinity is just a symbol, its actual nature is not relevant. In the same way that there exists essentially one normal sphere. The radiuses may differ, but any sphere can be deformed gently into another sphere, without tearing the surface. In exactly the same way a sphere is the same as the surface of a cube, but not with the surfce of a donut.
You can't say if any portion of the Riemann sphere appears Euclidean, or whether it is infinite in size or not. That because there is no concept of distance and size on the Riemann sphere. Any portion of the sphere can be stretched/shrank in any way as long as the sphere does not burst or separate patches merge.
The Riemann sphere does not get applied directly much beyond math, or otherwise I never heard of it. It is a useful construct, but rather abstract.
Above I talked about the topology of the Riemann sphere, not its differential geometry . But that would be harder to explain.
I don't know how satisfactory you found the answers. Try to read them though, and let me know if you have questions. Oleg Alexandrov ( talk) 02:37, 16 April 2006 (UTC)
Elroch made a lot of good edits to this article. It is now more mathematically correct, but it is hard to understand for somebody not knowing math however. I believe the geometric viewpoint, which, if not entirely accurate, was helpful in illustrating what is going on. Wonder what you think. Oleg Alexandrov ( talk) 15:09, 17 April 2006 (UTC)
This article (like the Stereographic projection article) prefers the projection onto a plane tangent at a pole over the projection onto a plane through the equator. It even asserts that the former is more popular. This contradicts my experience. Among the texts in front of me, the equatorial version is used books by Rudin, Bredon, Thorpe, Oprea, and Brown/Churchill; the pole-tangent version is used in Do Carmo; Spivak uses both versions.
More importantly, this article contradicts itself. Its claims about the transition maps being , corresponding to , etc. are all assuming the equatorial version. (In the equatorial version, the unit circle in is sent to the equator; in the pole-tangent version, the circle of radius 2 is sent there, so the unit circles from the two complex planes don't match up.)
In short, while the pole-tangent version of stereographic projection has uses in differential geometry, the equatorial version seems unequivocally better suited to complex analysis, which is where this article comes in. So unless there are objections I'm going to start making these changes. Joshua R. Davis 19:55, 25 April 2007 (UTC)
I appreciate that mathematical experts may be writing this article from their own minds, without referring to source material. I've done the same in articles where I have expertise. This is Wikipedia though, and at least a few citations are needed. At a minimum, cite a few books or papers that describe a Riemann sphere. Davidwr 15:10, 3 May 2007 (UTC)
Hello all. I just added applications and references to the article. But in fact I have rewritten the entire article. I think the new version is more precise/rigorous, clearer about what the metric is and what depends on the metric, better organized, less redundant, etc. (But then I would think that. :) I have already transplanted some of the purely geometric material to Stereographic projection (also newly rewritten).
Because there has been recent activity on this article and this talk page (moreover by editors whom I know and respect) I don't want to impose a complete rewrite on everyone out of the blue, so I'm posting it here for comments/incorporation. Joshua R. Davis 04:18, 4 May 2007 (UTC)
In mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as
well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted .
On a purely algebraic level, the complex numbers with an extra infinity element constitute the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one- dimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.
As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane . Let and be complex coordinates on . Identify the nonzero complex numbers with the nonzero complex numbers using the transition maps
Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere.
Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane.
In other words, (almost) every point in the Riemann sphere has both a value and a value, and the two values are related by . The point where should then have -value ""; in this sense, the origin of the -chart plays the role of "" in the -chart. Symmetrically, the origin of the -chart plays the role of with respect to the -chart.
Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified with . On the other hand, the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
The uniformization theorem, a central result in the classification of Riemann surfaces, states that the only simply-connected one-dimensional complex manifolds are the complex plane, the hyperbolic plane, and the Riemann sphere. Of these, the Riemann sphere is the only one that is closed ( compact and boundaryless).
The Riemann sphere can also be defined as the complex projective line. This is the subset of consisting of all pairs of complex numbers, not both zero, modulo the equivalence relation
for all nonzero complex numbers . The complex plane , with coordinate , can be mapped into the complex projective line by
Another copy of with coordinate can be mapped in by
These two complex charts cover the projective line. For nonzero the identifications
demonstrate that the transition maps are and , as above.
This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line or smooth conic in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere's automorphisms, later in this article.
The Riemann sphere can be visualized as the unit sphere in the three-dimensional real space . To this end, consider the stereographic projection from the unit sphere minus the point onto the plane , which we identify with the complex plane by . In Cartesian coordinates and spherical coordinates on the sphere (with the zenith and the azimuth), the projection is
Similarly, stereographic projection from onto the plane, identified with another copy of the complex plane by , is written
(The two complex planes are identified differently with the plane . An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.) The transition maps between -coordinates and -coordinates are obtained by composing one projection with the inverse of the other. They turn out to be and , as described above. Thus the unit sphere is diffeomorphic to the Riemann sphere.
Under this diffeomorphism, the unit circle in the -chart, the unit circle in the -chart, and the equator of the unit sphere are all identified. The unit disk is identified with the southern hemisphere , while the unit disk is identified with the northern hemisphere .
A Riemann surface does not come equipped with any particular Riemannian metric. However, the complex structure of the Riemann surface does uniquely determine a metric up to conformal equivalence. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positive smooth function.) Conversely, any metric on an oriented surface uniquely determines a complex structure, which depends on the metric only up to conformal equivalence. Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface.
Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with constant curvature in any given conformal class.
In the case of the Riemann sphere, the Gauss-Bonnet theorem implies that a constant-curvature metric must have positive curvature K. It follows that the metric must be isometric to the sphere of radius in via stereographic projection.
In the -chart on the Riemann sphere, the metric with is given by
In real coordinates , the formula is
Up to a constant factor, this metric agrees with the standard Fubini-Study metric on complex projective space (of which the Riemann sphere is an example).
Conversely, let S denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on S. It follows that any metric on S is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since "roundness" is not an invariant of conformal geometry. The Riemann sphere is only a conformal manifold not a Riemannian manifold. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice.
The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form
where , , , and are complex numbers such that . Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.
The Möbius transformations are profitably viewed as transformations on the complex projective line. In projective coordinates, the transformation is written
Thus the Möbius transformations correspond to complex matrices with nonzero determinant. These are the projective linear transformations .
If one endows the Riemann sphere with the Fubini-Study metric, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of , namely . It is isomorphic to the rotation group , which is the isometry group of the unit sphere in .
Overall, I think this is a big improvement. It's very nicely organized and well written. My main criticism is with the section "As a sphere". This section should really start out with a description of the stereographic projection rather than its inverse. The complex coordinates should be given by
(which are incidentally much simpler and easier to grok than their inverses). This shows very geometrically how to put complex coordinates onto the sphere to make it into a complex manifold. One can then give the inverse transformations to complete the picture.
There is a subtlety that occurs in these equations which should be explained. When projecting from opposite poles, one is not quite projecting onto the same plane. The planes are complex conjugates of each other. This is necessary to make the transition functions holomorphic rather than antiholomorphic. I remember this confused me a good deal when I first encountered the Riemann sphere. I think it is also the reason I, at one time, preferred projections onto the polar tangent planes: as the planes were distinct it was less easy to confuse them.
Minor quips: I prefer using z and w for the coordinates rather than ξ and ζ (which, by the way, appear nearly identical with my fonts), but I can see how this might lead to confusion. Also, for reasons unbeknownst to me, it seems much more natural to complex conjugate the southern hemisphere chart (i.e. when projecting from the north pole) rather than the northern one so that
Maybe its because I live in the northern hemisphere and I'm biased. But again this is a personal preference and I'm not bound to it. -- Fropuff 08:11, 4 May 2007 (UTC)
Another thought regarding the metric: the way you've written it makes it sound like the round metric is chosen arbitrarily, which is, of course, far from the truth. Firstly, the metric is the pullback of the round metric on the sphere by the inverse stereographic projection. Secondly, the complex structure on the sphere uniquely determines a conformal class of metrics on the sphere. Within this class there is a unique (or so I recall) metric with constant curvature. Normalize the curvature to be +1 and you get the round metric. Perhaps it would be best to put all the stuff about the metric in its own section. -- Fropuff 08:43, 4 May 2007 (UTC)
If we want to stick with Greek letters for complex variables then ζ and ξ are fine, and like it or not, your choice of which plane to conjugate does seem more common in the literature so I guess I'll have to live with it. The only thing I really miss from the present article is a description of the stereographic projection in spherical coordinates. In present notation we should have
where θ is the zenith angle and φ is the azimuth angle. I find these formulas extremely useful. They show clearly that circles of latitude on the sphere map to circles on the plane while lines of longitude map to radial lines. I realize these formulas are at stereographic projection but its nice to see them in complex notation. Other than that I am happy to replace the article with the new version. -- Fropuff 00:18, 7 May 2007 (UTC)
Why do we have this image? What does it half to do with the Riemann sphere? It's relevence is never explained. As far as I can tell, only the title of the image has anything to do with the article. The Riemann sphere is topologically a sphere, but has no one intrinsic geometry (you can get a metric by pulling back a metric with stereographic projection, but even then there is no unique choice of projection). Loxodromes are intrinsically geometric objects, and so would seem to have little to do with the Riemann sphere. And even if you do give the Riemann sphere a metric, the obvious choice would be a pullback via stereographic projection, and even then a loxodrome wouldn't come from any natural curve on the complex plane. The curve that maps to a loxodrome would be a "spiral" starting out at the origin and winding around finitely many times before shooting out to infinity (as opposed to natural spirals, which wrap around infinitely often). No natural function does that. So the image strikes me as a complete nonsequiter, and even if it isn't, it needs to be tied into the article somehow. skeptical scientist ( talk) 12:41, 1 July 2007 (UTC)
I cannot deduce from the formulaes in the article how to transform a point P=x+i·y of the Gaussian plane onto the x,y,z Cartesian coordinates (or longitude,latitude spherical coordinateS) of the Riemann sphere. :-/ Is it that complicated? -- RokerHRO ( talk) 09:50, 18 May 2010 (UTC)
∞ + ∞ = ∞ is stated at the section Arithmetic operations. I think this operation must be moved to the undefined operations (together with ∞ - ∞ and 0·∞). You can find a counterexample at page 30 here. I'm not an expert, so I'm afraid I may not be correct and I didn't change it. So please, if someone agrees, change it.-- Ssola ( talk) 17:57, 7 March 2012 (UTC)
This article is to dispell the obvious mistake made in the Riemann sphere definition which states that "∞" infinity is near to very large numbers and that zero "0" is near to very small numbers. This is false for two reasons: 1) nothing and infinity are unlimited concepts and are therefore by their very nature completely incompatible mathematically and conceptually with limited numbers. 2) The concept that zero is close to small numbers and infinity is close to large numbers is based on the assumants inability to conceive of something infinite outside of space and time resulting in the assumants mind counting up or down forever caught in a limited process that given enough time would never reach its infinite goal for the simple fact that it is always limited.
Conclusion this theory is logically refuted as false. — Preceding unsigned comment added by 193.200.145.253 ( talk) 10:52, 23 July 2013 (UTC)
The article addresses the problem of division by zero and rules out the special cases ∞ + ∞, ∞ - ∞ and 0 ⋅ ∞ as undefined. But what about 0/0? Is it still left undefined, as it is with the real numbers? My intuition tells me it would... SBareSSomErMig ( talk) 07:04, 25 September 2013 (UTC)
Why does this article say infinity plus infinity is undefined?? It has always made sense to define it as infinity. Infinity plus negative infinity is undefined though. Georgia guy ( talk) 22:00, 16 January 2014 (UTC)
Some parts of this article seem to imply any stereographic projection of the complex plane to a sphere is "a Riemann sphere", whereas other parts seem to imply there is only one canonical projection which forms "the Riemann sphere". Some consistency would be nice here.
My intuition from elsewhere in mathematics is that the former is true, even though the wording of the latter is more likely (e.g. in computational complexity we may refer to "the first Turing machine enumerated in shortlex form as a member of {0, 1}*", ignoring the fact that which Turing machine this may be depends entirely upon the coding language being used to model the machine in its coded form--generally because it has been proven that the choice of coding method is immaterial as long as all possible machines can be enumerated). TricksterWolf ( talk) 03:31, 6 February 2014 (UTC)
Any two stereographic projections give the same complex structure on the sphere (and also the same conformal structure). So if one thinks of the Riemann sphere as a complex or conformal manifold only, then there is no preferred projection. However, if one thinks of the Riemann sphere as the extended complex plane, then there is a preferred point at infinity and a preferred stereographic projection. Which of these is intended depends on the context. (Ironically, one of the participants in this very discussion has a username that illustrates both of these cases, with g=0 and different values of n.) Sławomir Biały ( talk) 16:33, 6 August 2014 (UTC)
It is claimed in the Rational functions section that "the set of complex rational functions ... form all possible holomorphic functions from the Riemann sphere to itself". As far as I know, is not a rational function, yet it is holomorphic. So this claim appears to be false. The only way out I can see is if we consider to be a "polynomial" on the grounds that it has a power series, but this seems to be a stretch. Am I mistaken? Luqui ( talk) 05:19, 8 July 2018 (UTC)
It seems to be that the topic of the extended complex plane is a bit primary to the concept of the “Riemann sphere” per se, and subordinating the former to the latter seems a bit backward.
We might start by defining the extended complex plane as (a) the complex numbers plus an extra point at infinity with defined topologically to be the center of balls , but it could alternately be defined as (b) equivalence classes of pairs of complex numbers such that iff i.e. the ratios or (c) some atlas of charts, e.g. two separate stereographic projections of the sphere from opposite poles onto the complex plane.
Then a section about the “Riemann sphere” (which should perhaps better be named after Carl Neumann, Riemann's student who came up with / promoted the idea) could discuss using chord length or arclength (from inverse stereographic projection) as a distance function between arbitrary points in the extended complex plane, and could discuss the use of complex numbers as a sometimes convenient representation for studying spherical geometry.
But the notion of the extended complex numbers doesn't seem to inherently require thinking about a "sphere" or spherical distance. The Möbius transformations of the extended complex plane are a broader group than only the isometries (rotations) of the sphere, and also include translations of the plane ("parametric transformations"), scaling ("hyperbolic transformations")
– jacobolus (t) 23:33, 24 January 2023 (UTC)
On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states that every simply-connected Riemann surface is biholomorphic to the complex plane, the hyperbolic plane, or the Riemann sphere.It seems to me like any article about the extended complex numbers should be talking extensively about how as a number system they are commonly used to model points the 2-sphere, Euclidean plane, and hyperbolic plane (for better or worse; arguably there’s at least a bit of conceptual mismatch). But in an article entitled “Riemann sphere” those don’t quite seem to belong. – jacobolus (t) 06:05, 25 January 2023 (UTC)
The section As a sphere contains this paragraph:
"In order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point and the second except the point . Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back-to-back at . Note that the two complex planes are identified differently with the plane . An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic."
This paragraph cries out to be rewritten clearly and accurately.
Already its first sentence describes stereographic projection exactly backwards.
Nobody has any idea what "intuitively seen as glued back-to-back" means.
In the phrase "... the two complex planes are identified differently with the plane", it is entirely unclear what "the plane" refers to.
And in the comment about "orientation reversal", there is no mention of what it is that has its orientation reversed, or where complex conjugation enters in.
(Is there really anything that gets its orientation reversed? Seems extremely doubtful.)
One of the worst paragraphs I've seen in a Wikipedia mathematics article.
— Preceding unsigned comment added by 2601:200:c082:2ea0:7185:a86f:eb51:745a ( talk) 11:38, 29 May 2023 (UTC)
To split or not to split? The page deals with a particular one-dimensional complex manifold known either by the stereographic projection of the page title or as the complex projective line. Every simply connected, one-dimensional complex manifold is isomorphic to this one, or the included plane, or the disk. On the side of sphere there is the realm of Riemann surfaces as a superset, whereas on the side of the projective line there is projective geometry. The combination of these two branches in one article corresponds an expectation that readers see that the two branches stem from the same trunk. — Rgdboer ( talk) 01:43, 28 October 2023 (UTC)