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Is the Riemann sphere anything more than a simple visual tool to help students understand the complex planes, or the behavior of complex valued functions at infinity, limit points etc?

Or is there a practical use in calculations of complex valued functions using the topology or geometry of the the sphere itself?

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  • $\begingroup$ The Riemann sphere is a (one point) compactification of the complex plane and is the simplest Riemann surface. Both of these facts lead to quite rich areas of study. $\endgroup$ – Michael Burr Aug 19 '15 at 20:00
  • $\begingroup$ No, it's very important as a Riemann surface and manifold, rather than just as a teaching tool for students or visual analogy. Now phasors, on the other hand, are quite silly. :) $\endgroup$ – anomaly Aug 19 '15 at 20:03
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    $\begingroup$ one practical use is the pull-back of the spherical metric to the plane to give it the non-Euclidean spherical geometry. I think that is a neat use of the Riemann Sphere. $\endgroup$ – James S. Cook Aug 19 '15 at 20:04
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    $\begingroup$ In addition, it is the projective space $\mathbb{P}_\mathbb{C}$, which is important in algebraic geometry. $\endgroup$ – Michael Burr Aug 19 '15 at 20:04
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The Riemann sphere is an essential object, and is certainly no mere didactic tool. For one thing the Riemann sphere can be given the structure of a complex manifold by using the maps $z \to z, z \to 1/z$ as charts from $\mathbb{C} \to S^2$. Then meromorphic functions on a domain $\Omega$ are simply the functions $f: \Omega \to S^2$ that are complex differentiable with respect to this structure.

Another example of a use of the properties of the riemann sphere is the following: take a map $f: \mathbb{C} \to \mathbb{C}$ that is bounded and holomorphic. Because $f$ is bounded, we can include $\mathbb{C} \to S^2$ by $z \to z$ and miss only $\infty$. But because $f$ is bounded we know that the singularity at $\infty$ is removable, and so obviously (alternatively by what is sometimes called "Riemann's Removable Singularities theorem") we can define $f$ at $\infty$ in a way that makes the resulting function $ \hat{f}: S^2 \to \mathbb{C}$ complex differentiable. Now $S^2$ is compact and so $\hat{f}(S^2)$ is compact and thus closed, but a non-constant holomorphic map must be open, which would imply that $\hat{f}(S^2)$ is either $\mathbb{C}$, which is absurd, or $\emptyset$, also ridiculous. Thus $\hat{f}$ must be constant and thus so was $f$.

This gives a cute proof of liouville that has a bit more topological flavor than it usually does. However you'll find that, upon unpacking the statements used, the proof reveals itself to be more or less the same ideas, just packaged in a different way.

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  • $\begingroup$ That was cool i never thought of it that way =) $\endgroup$ – T. Poindexter Aug 20 '15 at 3:13
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The Riemann sphere is not just a visual aid or an intuitive heuristic, it is a bona fide mathematical object that exists in its own right. For example, the uniformization theorem states the three simply connected Riemann surfaces are the open unit disk, the complex plane and the Riemann sphere.

The definition of Riemann surfaces mimics the definition of abstract smooth surfaces but using transition maps that are holomorphic instead of merely smooth, so it is a natural thing to do. In this context, meromorphic functions to the complex plane are holomorphic functions to the Riemann sphere. Topologically, the Riemann sphere is indeed a sphere, so the term is justified. Indeed it is the one-point compactification of the complex plane.

The Riemann sphere is also the natural context in which to study Mobius transformations, also known as linear fractional transformations. Consider the complex space $\Bbb C^2$. In order to obtain the complex projective line $\Bbb P^1(\Bbb C)$ we must mod out the action of $\Bbb C^\times$. All of the elements of this projective line, written in homogeneous coordinates, look like $[z,1]$ with $z\in\Bbb C$ with the exception of the point $[1,0]$, which you can interpret as the "point at infinity" adjoined to the plane $\Bbb C$. This is one way to define the Riemann sphere. The induced action of ${\rm PGL}_2(\Bbb C)$ can be expressed by the rule $(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})z:=\frac{az+b}{cz+d}$ subject to the conventions that dividing by $0$ yields $\infty$ and conversely.

In the Riemann surface setting, these Mobius transformations are precisely the automorphisms (so, conformal self-maps) of the Riemann sphere.

From the perspective of symmetry and symmetry groups, since these maps act transitively on the sphere (indeed, on its tangent bundle), it is a homogeneous space, which means the space looks "the same" from every point in the space, just as with Euclidean space where you can apply translations/shifts to move any point to any other point and apply rotations to relate any tangent vector to any other tangent vector. In this way, intrinsically speaking, no point is privileged or more special than any other point (this is not the case when considering arithmetic operations, of course), so excluding the point $\infty$ would be unnatural from this vantage point!

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    $\begingroup$ (This doesn't really answer the question about practicality. I think pretty much anything you can do for functions of the plane using the sphere can be done without the sphere, but perhaps not as elegantly. JSempliner's example of Liouville's Thm is nice.) $\endgroup$ – whacka Aug 19 '15 at 20:35
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Initially, the sphere $\mathbb{S}^2$ is just a set of points. Sets are okay, but kind of boring.

But we can go further: we can endow the sphere with a topology to make it into a topological space. The standard way of doing this makes the sphere into a compact surface, which is nice.

We can go further still: having given the sphere its usual topology, we can endow it with the structure of a Riemann surface. In other words, it is a surface (that is not $\mathbb{C}$) on which we can do complex analysis.

It is kind of cool that we can do complex analysis on surfaces other than $\mathbb{C}$. One could ask, for instance, whether the complex analysis theorems which are true on $\mathbb{C}$ carry over to $\mathbb{S}^2$, or whether one can prove theorems on $\mathbb{C}$ by looking at $\mathbb{S}^2$. Other users have given answers that illustrate such ideas.

Point: The Riemann sphere is the sphere with extra structure. That is, "Riemann sphere" is the name that we give to the sphere with its usual topology and Riemann surface structure.

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  • $\begingroup$ Fantastic answer as well. I've been looking for a reason to study more topology. =) $\endgroup$ – T. Poindexter Aug 20 '15 at 3:23
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If the Riemann sphere is merely a "simple visual tool", then equally the Argand plane is a "simple visual tool". That is, writing an imaginary number $z$ as $x+iy$ where $x$ and $y$ are real.

Jean-Robert Argand was an amateur who made a valuable contribution to mathematics.

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