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

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Aristotle's Axiom in Hyperbolic Geometry

I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...
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1answer
15 views

Freeness of the group generated by two hyperbolic isometries

If $f$ and $g$ are two hyperbolic isometries of the hyperbolic space $\mathcal H^n$, we know that $f$ has 2 fixed points on $\partial \mathcal H^n$ and similiarly for $g$. Is it true that $f$ and $g$ ...
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1answer
29 views

Cosine Law Duality in Hyperbolic Trigonometry

From setting up a hyperbolic triangle with hyperbolic side length $a,b,c$ and corresponding angles $A,B,C$, it is not hard to prove the following law of cosine: $$\cos A= \frac{\cosh b \cosh c -\...
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0answers
26 views

Formula for curvature of a hyperbolic plane

I would like to understand the curvature of a hyperbolic plane better in relation to the underlying Euclidean model and intrinsically without a model. I only consider the Beltrami-Klein model and the ...
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2answers
30 views

Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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1answer
23 views

Curvature of a hyperbolic plane

Consider a projective plane and a real quadric. According to the Klein-Beltrami-model the inside of the quadric is a hyperbolic plane. Klein proved that this plane has a constant negative curvature. ...
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2answers
33 views

Confusion about Hyperbolic Geometry.

I was introduced the Poincaré Disc model of hyperbolic geometry. The concept idea was clear but I had some questions about it that I could not figure out myself. I understand that the geometric ...
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0answers
12 views

Proof verification: Cross Ratio

Prove: If $[z_1,z_2,z_3,z_4] \in \mathbb{R} \cup \{\infty \} $, then $z_1,z_2,z_3,z_4$ are either concyclic or collinear. My proof below uses the geometric interpretation of cross ratio. I am not ...
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17 views

Non integer, non-centered Gaussian moments

I have read the following question : Non-centered Gaussian moments where it is stated that : $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{...
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1answer
44 views

What are the hyperbolic rotation matrices in 3 and 4 dimensions?

So the hyperbola-preserving transformation in 2 dimensional space is given by the matrix \begin{pmatrix} \cosh(\phi) & \sinh(\phi) \\ \sinh(\phi) & \cosh(\phi) \end{pmatrix} I'm wondering ...
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32 views

homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
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12answers
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What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
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1answer
52 views

Fundamental group of the unit tangent bundle on the genus 2 torus?

I'm interested in the 3-dimensional model geometries; specifically $\widetilde{SL}(2,\mathbb{R})$. I'm looking for a good (see, easily visualizable) example of a compact manifold formed as a quotient ...
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0answers
78 views

Studying the hyperboloid model, what is represented by the conic sections?

I am trying to get my head around the hyperboloid model of hyperbolic geometry. Hyperboloid_model The article is much too technical for me, please improve. And was thinking the hyperboloid $ x^2 +...
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1answer
121 views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be ...
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1answer
50 views

Intuition Behind the Hyperbolic Sine and Hyperbolic Cosine Functions

After enough time studying mathematics, we develop an instinct for the sine and cosine functions and their relationship to our standard Euclidean Geometry. I have come across the functions $\sinh(x)$ ...
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0answers
51 views

Area of surface of revolution between planes/concentric cylinders of Sphere/Pseudosphere

If area of surface of revolution with maximum radius $R,$ between two concentric cylinders radii $a,b$ is $$ 2 \pi R (a-b), \tag 1$$ then find equation of its meridian. EDIT2: i.e., find r(z) ...
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14 views

Does 3 points on Poincaré Disk geodesic lie on the same Poincaré Half Plane geodesic?

This may be a trivial question, IMO the answer should be yes. Given a geodesic $\delta$ on the Poincaré Disk's model with $A, B, C \in \delta$ And given that $f(x)$ is an isometry from the Poincaré ...
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1answer
32 views

Equidistant curves in the Half-Plane model.

Definition: An equidistant curve can be one of the three following: A hyperbolic circle, a horocycle or an equidistant line. In the Half-Plane model, a hyperbolic circle is represented by an euclidian ...
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5answers
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Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
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27 views

Definition of hyperbolic lenght.

Theorem 1: Let $\text{arc(AB)}$ be an arc of an equidistant curve (Which can be a circle, a horocircle or an equidistant line) and $(A^{n})$ a sequence of partitions of the arc $\text{arc(AB)}$ such ...
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1answer
29 views

In hyperbolic geometry, exactly how big is a dodecahedron composed entirely of right angles? [closed]

Specifically, I need to know the distance from the center to the vertices, and the distance to the faces.
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1answer
37 views

Metric Matrix of the hyperbolic Riemannian manifold

Let $\Bbb{H}^n:=\left\{(x_1,...,x_n)\in\Bbb{R}^n\mid x_n>0\right\}$ be the hyperbolic space and $g={d^2x_1+\dots+d^2x_n \over x_n^2}$ be the standard hyperbolic metric. Looking at the $\left(\Bbb{...
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1answer
33 views

Poincaré cylinder

The Poincaré disk model of the hyperbolic plane is the open disk ${\rm int}(D^2)$ with a certain metric $d_H(x,y)$. What happens if I take the open tube ${\rm int}(D^2)\times\Bbb R$ with the metric: $...
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26 views

Geometry - Inversion/Cross Ratios

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The ...
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1answer
26 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
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1answer
20 views

Hyperbolic distance of a point from center in Klein-Beltrami disk model

According to the Wikipedia entry about Klein Beltrami disk, I found that the hyperbolic distance between two points P and Q is determined by the following formula : $$d(P, Q) = \frac{1}{2} \ln \frac{|...
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1answer
29 views

Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
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27 views

How to compute a distance in the hyperbolic plane.

I know that a hyperbolic distance is defined by: $d_{\mathbb{D^2}} = 0 $ if $z_1 = z_2$ and $- \log[z_1, z_2, z_1^{\infty}, z_2^{\infty}]$ We define $\mathbb{D^2} = \{z \in \mathbb{C} $ such that $\...
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Showing that the inversion of a circle can be written in a certain way

Let $\phi = \dfrac{1}{r} ∘ T_{-O_C}$, $O_C$ the center of $C$, $T_{-O_C}$ is a translation. I want to show that the inversion of a circle $C \in \mathbb{C}$ can be written as: $$\iota_C = \phi ∘ \...
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Area of polygon of hyperbolic disc

Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex. {hyperbolic disc} There exists formula which defines ...
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2answers
35 views

Distance in Poincaré disk from origin to a point given

Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$ The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the ...
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4answers
174 views

Element of infinite order for a given group presentation

Let $G=\langle a,b,c,d \mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle$ be our presentation. The claim is that the commutator $[a,b]$ has inifinite order in $G$. I think this might be related to small ...
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27 views

Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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1answer
57 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...
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58 views

Hyperbolic plane shrinking

A very small area of the hyperbolic plane looks more Euclidean as the curvature approachs 0. Any more evidence? Or reference would help? Thanks
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22 views

Does the Ray Casting Algorithm works in Poincare's Disk to detect if point is inside Polygone?

As the Ray Casting Algorithm looks to me like a geometric construction on geodesics, and geodesics are redefined in Poincare's Disk, I feel this method would also work in hyperbolic geometry. Is this ...
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1answer
61 views

Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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28 views

Seeking proof to a Hyperbolic polygon conjecture

In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been ...
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1answer
34 views

Simple proof of the existence of lines in the hyperbolic space

Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$ g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}. $$ What is the easiest way to show that there exist at least one ...
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1answer
46 views

Geodesic curvature of a curve in the hyperbolic plane

Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$ Calculate the geodesic curvature of $\gamma$. The problem I'm ...
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0answers
28 views

Dirchlet region for the Hecke Triangle group

Let $G_n$ for $n>2$ be the subgroup of $SL_2(\Bbb R)$ generated by $$ \begin{bmatrix} 0 & -1\\ 1 & 0 \\ \end{bmatrix} \ \text{and} \ \begin{...
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35 views

curvature of planar hyperbola

If a planar hyperbola is parametrized as follows $$ x(t) = a\sec(t), \quad y(t) = b\tan(t), \quad a, b \text{ constants} $$ what is the curvature of the hyperbola curve?
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Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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1answer
49 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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22 views

On the connection between Bloch's space semi-norm and Bergman's hyperbolic metric.

On the proof of the following theorem $f\in \mathcal B \Leftrightarrow \beta(f)=\sup\left\lbrace\dfrac{|f(z)-f(w)|}{d_{\mathbb D}(z,w)}:z,w\in \mathbb D, z\neq w\right\rbrace$, where $\mathcal B$ is ...
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1answer
34 views

Calculate the midpoint of the line given in the Beltrami-Klein model

I'm given that the distance between two points in the Beltrami-Klein model is $$d(XY)=\frac{1}{2}ln\Big(\frac{\overline{XQ}\cdot\overline{YP}}{\overline{XP}\cdot \overline{YQ}}\Big)$$ where $P$ and $Q$...
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66 views

Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article Intrinsic Metrics on ...
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20 views

Triangle on Beltrami pseudosphere with angle sum $180^\circ$

What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?
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27 views

Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation?

I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where $$...