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Questions tagged [riemann-surfaces]

For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

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Is the Riemann surface plot in Wolfram Alpha wrong?

I'm taking the complex variable course and I find the Wolfram Alpha website very helpful for me to understand the Riemann surface. For example typing "Riemann surface z^(1/2)" I get nice plots but when I try (z+1)^(1/2)+(z-1)^(1/2), I get something…
xslittlegrass
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Definition of the ideal boundary in the context of Riemann surfaces

What is the ideal boundary of a Riemann surface? I came across this in 'A Primer on Riemann Surfaces' by Beardon. On pg.152 it is stated that "a compact surface has no ideal boundary, a parabolic surface has a 'small' ideal boundary, and a…
George
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Minkowski Norm is Euclidean if and only if the mean cartan torsion is = 0

Can any one give me the proof of; Minkowski Norm is Euclidean if and only if the mean carton torsion is = 0 ?
SAB
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Computing order of meromorphic function on hyperelliptic Riemann surface

Let $X$ be a hyperelliptic Riemann surface defined by the zero set of the function $f(x,y) = y^2 - x^5 +x$. I would like to compute the principal divisors of the meromorphic function $x$ and $y$. In general, if $g$ is a meromorphic function on some…
warzasch
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On the transitivity of the group of automorphisms of a Riemann surface

Let $S$ be a Riemann surface. What can be said of the greatest integer $n$ such that the group of biholomorphisms of $S$, $\mathrm{Aut}(S)$, acts $n$-transitively on $S$ ? (for the Riemann sphere, it is 3 for instance) In particular, is there any…
Albert
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Meromorphic 1-form definition question.

Let $X$ be the Riemann sphere with local coordinates $z$ in one chart and $w=\frac{1}{z}$ in the other. Let $\omega$ be a meromorphic 1-form on $X$. Show if $\omega = f(z)$d$z$ in the coordinate $z$, then $f$ must be a rational function of $z$. So I…
homosapien
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Defining the Riemann sphere in terms of the complex projective line corresponding to $\mathbb{C}^2$

Out of curiosity I have started a tutorial in Riemann surfaces. But since I am mostly trained in (stochastic) analysis a lot of the prelimenaries from differential geometry and algebraic geometry are not always clear or known to me. (So you can…
Aris
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Riemann surfaces from weierstrass's idea (as a complete analytic function)

Currently I am reading some textbooks on Riemann surfaces, mainly Donaldson's book. In Donaldson's book (and some other quite newly wrote book), most Riemann surfaces seem constructed in a "algebraic curve" style or some quotient space style, which…
onRiv
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Functions on a compact Riemann surface that take zero for every principal divisor

Let $X$ be a compact Riemann surface and $u: X \longrightarrow \mathbb{R}$ be a smooth real-valued function. If $u(\operatorname{div}(f))=0$ for every meromorphic function $f$ on $X$, does this imply that $u$ is constant? Here let $D=\sum n_i P_i$…
QU Binggang
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Does there exist a non-constant holomorphic function on any open Riemann surface?

This question is motivated by the following theorem: "If $X$ is a Riemann surface (connected, Hausdorff, holomorphic charts), and there exists $f: X \to \mathbb{C}$ non-constant, then $X$ is second-countable." This statement feels…
Colin Fan
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Inverse of holomorphic map on Riemann surface is holomorphic?

Let $X,Y$ be Riemann surfaces and $f:X→Y$ is holomorphic map between $X$ and $Y$ and $f$ is bijective. Then, could you tell me the proof of the inverse of $f$ is holomorphic map? I proved this in special cases but I'm not sure where this holds in…
user925204
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Riemann Surface of $z^2$

I understand (I think) how the Riemann surface of $\sqrt z$ is constructed, or even that of $\log z$. I can visualize them by transforming a plane according to the inverse of the function, i.e. $z^2$ for the former and $\exp z$ for the latter. See,…
Eser Aygün
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Impossible Construction of Riemann Surfaces à la Hurwitz

I'm currently trying to construct simple Riemann Surfaces in the way of Hurwitz (see e.g. here): Given the complex plane $E$, which originates at $O$ and $w$ non-identical points $a_k$. Cut the plane from $O$ to each $a_k$, which shall be denoted…
draks ...
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Genus of an open Riemann surface, definition and meaning?

How is the genus of an open Riemann surface $X$ defined? How does it relate to the genus of a compact Riemann surface into which $X$ embeds as open subset? How can the genus of $X$ be computed as an intrinsic invariant? I read the paper of Pallete…
Jo Wehler
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Showing something is not a Riemann surface

So suppose we have $X=\{(x,y)\in \mathbb{C} : y^2=x^2+x^3\}$ and i am asked to see that this is not a Riemann surface. Well we know that it is not going to be a smoth affine plane curve because at the point $(0,0)$ both the derivatives are $0$ but…
Someone
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