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

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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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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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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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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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Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $. at the ...
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33 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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Find the hyperbolic length of the geodesic segment

I'm reading my textbook and I'm trying to make sense of this example. So the place with the red star shows the actual process of calculating the hyperbolic length. My question is how they get the ...
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64 views

We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...
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119 views

Comparing two fundamental domains for $\Gamma(2)$

(0). My question concerns the relation between two different fundamental domains for the group $$ \Gamma(2)= \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}\in SL_2(\mathbb Z) \; ...
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118 views

Mapping a line in the hyperboloid model of $\mathbb{H}^2$ to a circle in the Poincaré model

In the hyperboloid model of $\mathbb{H}^2$ a point $P(x,y,z)$ is the intersection of the vector $(x,y,z)$ with the upper sheet of the hyperboloid $x^2 + y^2 - z^2 = -1$, and a line $L(a,b,c)$ is the ...
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48 views

Side and angle relations in a hyperbolic quadrilateral.

Let $PQRS$ be hyperbolic quadrilateral, i.e. a quadrilateral in $\mathbb{H}$ whose sides are hyperbolic geodesic. Let length$(PQ)=l_1$, and length$(PS)=l_2.$ Also $\angle SPQ=\theta_1$, $\angle ...
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167 views

measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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115 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
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103 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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Does a lattice in $PSL(2,\mathbb{R})$ stabilizing $\infty$ have a domain with vertex at $\infty$?

Suppose $\Gamma$ is a lattice in $PSL(2, \mathbb{R})$ acting on the upper half plane. Suppose that the stabilizer in $\Gamma$ of the point at infinity is nontrivial. Does it then follow that the ...
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133 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
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103 views

References for Hyperbolic Graph Theory

I'm sorry to disturb you but I really got stuck! I can't find any clear and, somewhat, complete reference for this topic. I'm looking for a book, or review, or survey or course notes regarding ...
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Progression of a point along geodesics under the action of hyperbolic Möbius transformations

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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353 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
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194 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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194 views

what is the use of the hyperboloid model for hyperbolic geometry?

I am quite new to hyperbolic geometry so even an answer that this question doesn't make any sense can be very helpful. As far as i understand: There are different models of a plane where hyperbolic ...
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Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess). First of all some definitions: Definition 1 (möbius-transformation): Let $A \in \mathbb{C}^{2 ...
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JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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232 views

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
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425 views

Calculating hyperbolic distance between two points

I was looking for a formula to calculate the hyperbolic distance between two planes and came across this formula $$\ln\left(\csc b-\cot b\over \csc a - \cot a\right)$$ where $a$ and $b$ are the ...
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41 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
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Is there non-discrete group isomorphic to the fundamental group, what about the quotient?

It is known that (uniformization theorem) any Riemann surface can be written as the quotient of its universal cover by a discrete group (of Möbius transformations). This group is isomorphic to the ...
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Relations between Kleinian groups and quotient manifolds

In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of ...
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The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve. Let $P$ be the identity element of $E$. Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
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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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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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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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What is the geometric center and what is the other point?

In Euclidean geometry it is simple: In a triangle $\triangle ABC$ there is a single point $H_a$ on $BC$ such that the triangles $\triangle ABH_a$ and $\triangle ACH_a$ have the same area. the ...
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Triangle Inequality for Hyperbolic Metric (Logarithm of Cross-Ratio)

I need clarification in one step of this answer to my previous question. I was re-reading it, and it isn't clear to my why we can make the assumption of $ \Im(p)< \Im(q)< \Im(r) $. (Immaginary ...
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Hyperbolic isometries preserve hyperbolic paralelism

In the Poicaré half-plane model, under Cayley-Klein metric, i. e., $ d:\mathbb{H}^2\rightarrow\mathbb{R}, d(A,B) = \big{|} log ...
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Formula for length of diagonal in a Lambert quadrilateral

Given a Lambert quadrilateral $AOBF$ where the angles $ \angle FAO , \angle AOB , \angle OBF $ are right, and $F$ is opposite $O , \angle AFB$ is the acute angle , and the Gaussian curvature = -1 (so ...
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The action of an S-arithmetic group on the hyerbolic plane

I have a really quick question. I am interested in $G=SL_2(\mathbb{Z}[1/p_1,...,1/p_n])$, where $p_1,..., p_n$ are prime numbers. Since $G$ is a subgroup of $SL_2(\mathbb{R})$, it acts in the ...
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uniquess of hyperbolic numbers

I'm trying to prove, the uniqueness of hyperbolic numbers, like the complex numbers are unique, but since hyperbolic numbers aren't a field, I can't use the ideas of this. Are there theorems of ...
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How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
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problem with proof 'horoball QI extension theorem'

I'm reading the book of Drutu and Kapovich "Lectures on geometric group theory". In the proof of Mostow rigidity theorem, they say that they can extend an $\rho$-equivariant function ...
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Hartshorne Exercise 41.2 - Altitudes of a triangle have a common perpendicular [hyperbolic]

This is my first post here ever, so don't be too rude, if i missed something. My question refers to exercise 41.2 "GEOMETRY:EUCLID AND BEYOND" from Robin Hartshorne. You can easily find the book as ...
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Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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Prove $A \cdot B\leq -1$, where $A$ and $B$ are in $\mathbb{H}^2$

Let $A$ and $B$ be in $\mathbb{H}^2$. I need to prove that the lorentzian dot product between $A$ and $B$ is less than or equal to $-1$. I have no idea where to start.
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sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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How to make a triangular pillow with a pre-drilled tube with two truncated tetrahedra? (from Jeff Weeks' paper)

In his paper Computation of Hyperbolic Structures in Knot Theory, p.12, Jeff Weeks explains as below how to make a triangular pillow with a pre-drilled tube by gluing two truncated tetrahedra. A ...
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233 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. ...