Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

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Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group?

Let $S$ be a compact R.S. admitting a Fuchsian model $\mathbb{H} / \Gamma$. We know that $\pi_1(S) \cong \Gamma$. Let $\mathcal{B} \subseteq S$ be a finite set of points, is $\pi_1(S - \mathcal{B})$ ...
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0answers
57 views

Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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1answer
26 views

The precise formula of the Poincare-Bergman metric on the disc $\mathbb{D}$.

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. If $\gamma:[0,1]\rightarrow{\mathbb{D}}$ is a $C^1$ curve in $\mathbb{D}$, we define the Bergman length of $\gamma$ by ...
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1answer
36 views

Algebraic solutions for Poincaré Disk arcs

Given two points on the Poincaré Disk, there is a single straight line or arc that passes through them and that is orthogonal to the unit circle. Using compass and straightedge methods, one can easily ...
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2answers
191 views

“Every geometry is a projective geometry!” So Hyperbolic geometry is a projective geometry?

The great mathematician Arthur Cayley (https://en.wikipedia.org/wiki/Arthur_Cayley ) seems to have said "all geometry is projective geometry" (sorry no exact source, probably it is somewhere in Felix ...
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38 views

Why does the halfplane model of the hyperbolic plane involve only the upper half of the plane?

The Hyperbolic metric $$ s= \pm \int \frac {\sqrt{1 +y'^2} \, dx}{y} $$ for geodesics in $ \mathbb H^2$ integrates to a full circle, but why only the upper half is considered? The query is about ...
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Applications of the Hurwitz Theorem on Number of Automorphisms?

So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem: Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic ...
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1answer
43 views

Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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1answer
52 views

Outer Automorphisms of PSL2(R)

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would ...
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Geodesics in a hyperbolic plane like space

For $|\rho| < 1$ and $\sigma >0$ consider the Riemannian metric \begin{equation} g:= \begin{pmatrix} \frac{1}{\left( 1-\rho^2 \right)y^2} & \frac{-\rho}{\sigma\left( 1-\rho^2 \right)y^2} \\ ...
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Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic 3-space. Let $TH$ be the tangent bundle of $H$. I have a question: Is $TH$ isometric to H times a flat $k$-space?
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1answer
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Triangle inequality of hyperbolic metric

For $z_1, z_2 \in \mathbb{B}^2$, define $d(z_1, z_2) = \text{cosh}^{-1}(1+ \dfrac{2|z_1 - z_2|^2}{(1-|z_1|^2)(1-|z_2|^2)})$. In my text book (Lee's Topological manifolds Problems 12-23), to prove ...
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2answers
158 views

Do minimal hyperbolic surfaces exist? What do they look like?

I understand that it is impossible to embed* the entire hyperbolic plane in $\mathbb{R}^3$. But, can one create a embedding of part of the hyperbolic plane such that the resulting surface is also ...
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1answer
21 views

What about the Fuchsian groups make them stand out?

Why do we stop at Fuchsian groups (I.e. discrete subgroups of automorphisms of the hyperbolic plane) when we study things like quotients and what not? Is there a maximalist or universality property ...
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Is the hyperbolic plane $\mathbb{H}^2$ special among the hyperbolic spaces $\mathbb{H}^n$? [closed]

Are there properties that hold for the plane but not for higher dimensional spaces?
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1answer
45 views

Is it true that a geodesic on a hyperbolic surface can be lifted to a geodesic on the hyperbolic plane?

Let $\mathbb{H}$ be the hyperbolic plane, $\Gamma < \text{Isom}(\mathbb{H})$ be a Fuchsian group, and $S = \mathbb{H}/\Gamma$. If $\gamma : [0,1] \rightarrow S$ is a geodesic, can it be lifted to a ...
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1answer
28 views

I am looking for an introduction to hyperbolic surfaces as a quotient of the upper half plane by lattices.

I keep coming across results of the form: If we take the quotient of the upper half plane by a Fuchsian group with this property, we get a surface with that property (cusps, funnels, in/finite area, ...
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1answer
158 views

Area of a right angled hyperbolic triangle as function of side lengths

I was puzzeling with Area of hyperbolic triangle definition and could not figure it out, but then i thought there should be a (maybe solvable) simpler problem so here it is: suppose: an ...
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1answer
69 views

Various definitions of Moduli space of Riemann surfaces and Uniformization theorem

I'm sorry for the quantity of questions I'm asking, but I would like to solve once and for all many doubts I have on equivalent definitions of the moduli space of Riemann surfaces. Definition 1: The ...
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1answer
24 views

A question on Fuchsian group and automorphism of surface

Let $\mathbb{H}$ be the upper half of the complex plane, i.e., $\mathbb{H} =\{ z \in \mathbb{C}: \operatorname{Im} z >0\}$. And let $V$ be a Riemann surface with genus $g \ge 2$. Then $V$ is ...
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1answer
73 views

Area of hyperbolic triangle definition

I found this question recently in my booklet on hyperbolic geometry asking a very simple question but I could not answer it: Why can we not define the area of a hyperbolic triangle as in the plane ...
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1answer
26 views

How is $PSL(2, \mathbb{R})$ explicitly identified with the unit tangent bundle $T^1(\mathbb{H})$?

Let's say I have a given matrix $\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$, what point and tangent direction is it identified with?
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1answer
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What is the relationship between the Liouvelle measure on $T^1(\mathbb{H})$, and the Haar measure $\mu$ on $PSL(2, \mathbb{R})$?

Let $\frac{dx dx d\theta}{y^2}$ be the Liouvelle measure on $T^1(\mathbb{H}) \cong \mathbb{H} \times \mathbb{S}^1$. Let $\mu$ be the Haar measure on $PSL(2, \mathbb{R})$. What is the relationship ...
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45 views

What are some good graphics programs for depicting hyperbolic geodesics?

I'm looking for some software that allows one to draw accurate pictures in hyperbolic space. In particular, I want to be able to specify pairs of points and generate the geodesic between them, in ...
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55 views

Difference between Euclidean and Hyperbolic lengths.

What is the difference between Euclidean and Hyperbolic lengths? For instance if i were to measure a curve on with the euclidean distance and alternatively the Hyperbolic distance, What would be the ...
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0answers
30 views

What is the metric of the Riemann surface resulting from quotiening the upper half plane by a Fuchsian group?

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. What is the metric we get on $\mathbb{H} / \Gamma$?
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214 views

What is the volume of sphere in hyperbolic space?

I'm looking for a formula to describe surface and volume of a sphere in hyperbolic 3-space. I found some results which were generalized for any dimension, but I wasn't able to understand them. ...
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1answer
132 views

Arc-length Parametrization of Geodesic in Hyperbolic Plane

Here's a related question. So, we have $H=\{(x,y)\in\mathbb{R}^2: y>0\}$ and define the metric $g=\frac{1}{y^2}(dx^2+dy^2)$. I know that the circle $x^2+y^2=1, y>0$ is the image of a geodesic in ...
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1answer
120 views

For this hyperbolic punctured torus, how does the Dirichlet domain change as we move its center?

Consider the Fuchsian group $\Gamma:=\Big\langle\begin{pmatrix}1&1\\1&2\end{pmatrix}, \begin{pmatrix}1&-1\\-1&2\end{pmatrix}\Big\rangle$. A commonly studied fact is that when this acts ...
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Examples of Supergroups: U(n | m), SU(N, n|m) and PSU(N, n|m).

Looking for explicit forms of group elements in the supergroups: (1) U(n | m), (2) SU(N, n|m), (3) PSU(N, n|m), (4) PSL(n|m), (5) OSp(n|m). We can simply take $N=2$, $n=1$ and $m=1$. Partial ...
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2answers
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Show a set is open in C

Show that the set O={z$\in\Bbb C$:Re(z)<0, Im(z)>0} is open in $\mathbb C$ I know this is the second quadrant, but I don't know how to do the proof.
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When $\cosh yx/2=\pm 1 $?

When $\cosh \frac{xy}{2}=\pm 1 $? is it correct to say $xy/2=cosh^{-1}(\pm1)$ Then $xy=2 \cosh^{-1}(\pm1)$ I think there is better solution for this problem? any idea?
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Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
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1answer
45 views

Poincaré disk model of hyperbolic plane

Can someone please explain trigonometry in Equations (2) to (8) of: PoincareDisk ?
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44 views

What are the most important issues to consider in upper-half plane model?

I hope you can help me, I need to do a research project about this model of hyperbolic geometry. Honestly, I've never studied the subject, and I'm not sure that subjects should give more importance. ...
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1answer
62 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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67 views

Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
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Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
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What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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Isometries of hyperbolic space

The metric tensor for the Poincaré ball model of hyperbolic geometry is $$ g_{ij} = \frac{\delta_{ij}}{(1 - \lvert \mathbf{r} \rvert^2)^2} $$ where $\mathbf{r}$ is the position in the ambient ...
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44 views

Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
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104 views

Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
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1answer
25 views

Why to include the $C$ in the formula for the distance in hyperbolic geometry?

I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would ...
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Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
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1answer
80 views

How (not) to plot a Hyperbola? [closed]

I am trying to plot a Hyperbola in Wolfram Alpha. Its giving me a strange graph. How to correct that?
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2answers
69 views

Error when computing geodesics in hyperbolic half plane

It is known that the geodesic equations for the upper half plane equipped with the hyperbolic metric are $$x''=\frac{2x'y'}{y},$$ $$y''=\frac{(y')^2 -(x')^2}{y}.$$ It is also well known that the ...
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1answer
52 views

Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) ...
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1answer
96 views

Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through ...
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1answer
238 views

Poincaré hyperbolic geodesics in half-plane and disc models

The objective of this post is to state that 1) the Poincaré hyperbolic metric results in a solution of complete geodesic circles in both half-plane and disk models. 2) the choice of one or other ...
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14 views

Cross Ratio of two rays through origin

There are two fixed and two variable concurrent rays of unit length in 3 space through the origin. How should the spherical coordinates of the two variable points be related to result in a constant ...