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Questions tagged [riemann-sphere]

For questions about the Riemann sphere, a model of the extended complex plane.

The set of extended complex numbers, or Riemann sphere, consists of the set $\mathbb{C}$ of complex numbers, together with a point $\infty$. This can be viewed as a sphere via stereographic projection from the north pole, with the pole itself identified with $\infty$.

From a topological viewpoint, the Riemann sphere is a one-point compactification of the space $\mathbb{C}$. In fact, the sphere can be viewed as a complex manifold with a well-defined complex structure.

It is known that the automorphisms of the Riemann sphere are precisely the Mobius transformations.

Reference: Riemann sphere.

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The Riemann Sphere Interpretation

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…
T. Poindexter
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Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with the standard chordal metric. The simplest functions…
pregunton
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Classifying sharply 3-transitive actions on spheres.

It is well known that mobius transformations act sharply $3$-transitively on the Riemann sphere (that is, any three points can be mapped to any other three points in a unique way). In this question, I asked generally about (continuous) sharply…
Beren Gunsolus
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Why is the complex domain of cosine naturally a sphere?

Near the end of this MAA piece about elliptic curves, the author explains why the complex domain of the cosine function is a sphere: since it's periodic, its domain can be taken as a cylinder, wrapping up the real axis. And because cosine of…
ziggurism
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Is infinity the reciprocal of zero/is zero the reciprocal of infinity?

Is infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero or infinity, for example) and you can't have a…
Rory M. Tims
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Roadmap to understand the link between spherical harmonics and Riemann sphere?

My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a crucial role in between since it connects Fourier…
Ooker
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A domain on a sphere is simply connected if and only if its complement is connected

I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to prove this without algebraic topology?
Spook
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When is $\infty$ a critical point of a rational function on the sphere?

In his "Iteration of Rational Functions" Beardon defines a critical point of a rational function (mapping the Riemann sphere to itself) as a point such that the function is not injective in any neighborhood of that point. I'm trying to square…
Aaron Golden
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Sharply $k$-transitive actions on spheres

A nice fact from complex analysis is that the mobius group acts sharply 3-transitively on the Riemann sphere. I am wondering if other sharply k-transitive (continuous) actions are known on any $S^n$, and if it's possible to classify them (perhaps up…
Beren Gunsolus
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What is the real and imaginary part of complex infinity?

I know that complex infinity is a pole on the Riemann-sphere, but what is actually the real and imaginary part of it? Is it $\pm\infty \pm \infty i$? And how is it different from simple non-complex infinity?
plasmacel
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Uniformization of metrics vs. uniformization of Riemann surfaces

The uniformization theorem in complex analysis says that T1. Any Riemann surface of genus $0$ is conformally equivalent to the unit sphere. The uniformization theorem in differential geometry says that T2. Any smooth Riemannian metric on $S^2$ is…
Alex Bogatskiy
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Definition of a pole in Riemann sphere

I'm working/familiar with the following defintion of a pole in $\mathbb{C}$: Let $D\subset \mathbb{C}$ be an open domain and $f:D\to\mathbb{C}$ a function. $z_0\in\mathbb{C}$ is called a pole of $f$, if $\vert f(z)\vert\to\infty$ for $z\to…
M.W.
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Characterizations of a linear fractional transformation

Consider the function $$ g(t) = \frac{1+it}{1-it} = \frac{1-t^2}{1+t^2} + i \frac{2t}{1+t^2}. $$ (The second equality holds except when $t=i$.) It seems to be widely known that this function is the only linear fractional transformation having these…
Michael Hardy
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Components of Riemann Tensor on sphere

Since the surface of a sphere is a 2D manifold, the Riemann Tensor should have only one component. But if I calculate according to $$R^i_{~rkj}= \frac{\partial\Gamma^i_{~jr}}{\partial u^k} -\frac{\partial\Gamma^i_{~kr}}{\partial…
Fuzzy
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What is the derivative of a polynomial at $\infty$?

Let $f$ be a polynomial defined on the Riemann sphere. I'm struggling to understand in what sense such a map can be said to be "holomorphic" at $\infty$. What is the derivative of $f$ at $\infty$? I have a chart $z\to\frac1z$ mapping $\infty$ to $0$…
Jack M
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