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

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1answer
15 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
22 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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0answers
24 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 $\...
1
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1answer
482 views

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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0answers
14 views

Metric Matrix of the hyperbolic reimmanian manifold

Let $\Bbb{H}^n:=\{(x_1,...,x_n)\in\Bbb{R}^n|x_n>0\}$ be the hyperbolic space and $g={d^2x_1+...+d^2x_n \over x_n^2}$ be the standard hyperbolic metric. Looking at the $(\Bbb{H}^n,g)$ remannian ...
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0answers
24 views

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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2answers
17 views

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
170 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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0answers
24 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}...
5
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1answer
54 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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1answer
55 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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1answer
18 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 ...
5
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1answer
54 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 ...
2
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0answers
26 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 ...
1
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1answer
30 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 ...
2
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1answer
42 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 ...
5
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0answers
28 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 ...
2
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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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0answers
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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36 views

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 ...
4
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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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0answers
20 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
33 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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0answers
61 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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1answer
19 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 $180^\circ$?
2
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1answer
25 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 $$...
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0answers
6 views

Metric relations in lambert quadrilateral

I already found the relations in a rectangle triangle (6 formulas for the sides) and for a general ordinary triangle (sine and cosine hyperbolic laws). But now I'm trying to find them for a triangle ...
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0answers
71 views

How low will a given string hang? [closed]

If I have a piece of string that is n meters long, attached at two points m meters apart, how low will the string hang? The two ...
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0answers
30 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
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0answers
29 views

Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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1answer
42 views

Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as $$K_\rho(z)=-\...
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0answers
18 views

Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
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1answer
358 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a Google search didn't give me much. Also, are ...
10
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2answers
200 views

Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
2
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1answer
32 views

Name of the modular group

I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's ...
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0answers
20 views

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

I am trying to get my head around the hyperboloid model of hyperboloic geometry https://en.wikipedia.org/wiki/Hyperboloid_model (article is much to technical please improve) And was thinking the ...
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0answers
14 views

Hyperbolic half-planes are geodesically-convex

I'm trying to understand the concept of Dirichlet domains associated to the action of a Fuchsian group $G$ on $\Bbb H$ (the upper half-plane of $\Bbb R^2$ endowed with its usual hyperbolic metric). ...
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0answers
29 views

Proof of $\delta$-Hyperbolicity of $\mathbb H^n$ just with the hyperboloid model?

Do you know any proof of the fact that $\mathbb H^n$ is Rips-hyperbolic (i.e., geodesic triangles are $\delta$-slim for some $\delta$, also called "Gromov-hyperbolic" in some contexts), which makes no ...
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1answer
43 views

Find the curvature of the metric $ds=\frac{|dz|}{(1+|z|^2)}$ on $\mathbb{C}$

The curvature of the metric $g$ is defined as $$k(z)=-\bigg(\frac{2}{\alpha(z)}\bigg)^2 \partial \bar\partial log \alpha(z)$$ where $\alpha$ is positive and real valued. Also $\partial=\frac{(\...
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2answers
46 views

The triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$

This answer made me wonder if there is a geometrical way to prove that the triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$. In other words, how can we ...
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1answer
16 views

Connected components of a subset of E

Let $E$ be a real vector space of dimension n+1 with a symmetric bilinear form B of signature (n,1). Let $H=\{x \in E : B(x,x) <0\}$. Somewhere I saw that it has two connected components. Can ...
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1answer
24 views

Geometrical interpretation of the conjugacy of triangle groups.

Let $\triangle$ and $\triangle'$ be two hyperbolic triangles of respective angles $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$. Let us suppose that the triangle subgroups $(\alpha,\beta,\gamma)$...
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2answers
34 views

Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$.

Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle ...
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1answer
34 views

Show every Mobius transformation $T(z)=\frac{\alpha z+ \beta}{\bar \beta z+ \bar \alpha}$ acts as an isometry of the hyperbolic disk

Consider the unit disk $\mathbb{D}=\{z: |Z| < 1\} \subset \mathbb{C}$ equipped with the hyperbolic metric $g$ induced by $1$ form $ds=\frac{|dz|}{(1-|z|^2)}$ I am trying to show that every Mobius ...
4
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1answer
221 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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1answer
48 views

How to graph in hyperbolic geometry?

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$....
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2answers
319 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...