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

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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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1answer
62 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
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2answers
151 views

Showing the function $f(x,y)$ is one by one

Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
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1answer
89 views

Projecting external points to a circle: Distance order preserving?

Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation: I compute the point of intersection of the i) circle and the ii) line joining each ...
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2answers
285 views

Simple non-closed geodesic.

In torus there exists simple non-closed geodesic. One example is take the irrational slope curves in $\mathbb{R}^2$ and project it down to torus. Is this thing can happen in closed hyperbolic surface ...
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1answer
109 views

Any hyperbolic $n$-simplex is contained in an ideal simplex

Recall that an $n$-simplex in $\overline{\mathbb{H}^n}$ (the closure of $n$-dim hyperbolic space) with vertices $v_0,...,v_n\in \overline{\mathbb{H}^n}$ is the closed subset of $\mathbb{H}^n$ bounded ...
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1answer
247 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
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2answers
157 views

topic for presenting in hyperbolic geometry

For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing ...
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105 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
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103 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
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1answer
94 views

Reference Request: Regge Symmetry “Angle-Edge” Duality

A tetrahedron in hyperbolic 3-space can be defined (up to isometry) by the measures of its dihedral angles, $(a, b, c, a^\prime, b^\prime, c^\prime)$, with $a$, $b$, $c$ along edges that meet at a ...
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147 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
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1answer
53 views

Questions on two elements in a Fuchsian group which have at least one common fixed point

This is a homework question I am unable to solve. Let $A,B \in PSL(2,R)=Aut(H)$. Assume none of them are elliptic and they have : Case 1) one common fixed point at the boundary of $H$ (i.e. they ...
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1answer
215 views

Set of points equidistant from two points in hyperbolic space.

Given two points $p,q$ in the hyperbolic plane, show that the set of points equidistant from $p$ and $q$ is a hyperbolic line. I am unsure how to proceed with this question. Would it be easier to use ...
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2answers
279 views

Reflection in a hyperbolic line formula

Let $H$ denote the upper half-plane model of hyperbolic space. If $L$ is the hyperbolic line in $H$ given by a Euclidean semicircle with centre $a\in \mathbb{R}$ and radius $r >0$, show that ...
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2answers
249 views

Measure on a quotient

Can anyone explain me the following : let $M$ be a hyperbolic manifold and $\Gamma = \Pi_1(M) \subset Iso(\mathbb{H}^n) $. How does the Haar measure on $Iso(\mathbb{H}^n) $ induces a measure on ...
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437 views

Möbius Transformations are Orientation Preserving?

This question is truly stupid, but is driving me crazy. I just need an outside viewpoint to sort out what's going on. In my textbook: "Show that every linear fractional (LF) transformation of ...
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1answer
365 views

Rigid body motion on the Poincare disc model of the hyperbolic plane

I'd like to implement an interactive simulation of an actor controlled by the user moving around in the Poincaré disc model of the hyperbolic plane. I need to know how to perform translation and ...
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31 views

Enlightening explanation of a theorem of Zimmert's

I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper: Zimmert, R. Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
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1answer
56 views

Regular triangulation of compact oriented hyperbolic space

Is there a good way of explicitly constructing a regular triangulation of a compact orientable hyperbolic 2-manifold, ideally with any desired vertex degree $\ge 7$? I only need the topology, not any ...
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1answer
204 views

Understand the Hyperbolic space

I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$. For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ But for $n>2$ I can't seem to ...
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164 views

Axis of the product of two loxodromic isometries

Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element. We consider the axes of these three elements. I'd like to know if ...
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148 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 ...
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2answers
782 views

What is the proof that rectangles do not exist in hyperbolic geometry?

I am in need of help figuring this out-- If the only straight lines in hyperbolic geometry are those that pass through the center, then isn't there a right angle? (horizontal and vertical) Which ...
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861 views

Construction of equilateral triangle in Poincare disc model

Points A and B are given in Poincare disc model. Construct equilateral triangle ABC. Any kind of help is welcome.
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110 views

Sum of angles in a hyperbolic triangle with one ideal angle

I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for ...
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1answer
87 views

Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
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1answer
93 views

Locally cyclic subgroups of a hyperbolic group

How can we show that locally cyclic subgroups (ie. groups whose finitely generated proper subgroups are cyclic) of a hyperbolic group are cyclic?
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477 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
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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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57 views

Projection on geodesic lines in $\mathbb{H}^n$

Good morning everyone, I was wondering wether or not is the projection on a geodesic line in $\mathbb{H}^n$ $1$-lipschitz for the hyperbolic distance. I asked myself this question because i ran ...
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reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result: For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
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103 views

Real tree and hyperbolicity

I seek a proof of the following result due to Tits: Theorem: A path-connected $0$-hyperbolic metric space is a real tree. Do you know any proof or reference?
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893 views

Books for Hyperbolic Geometry.

I want to read hyperbolic geometry. Can any one suggest some good books on the topic.
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1answer
134 views

Approximating Distances in the Hyperbolic Space

I am currently trying to understand the paper by Krioukov et. al. on hyperbolic networks, but since I do not have a background in hyperbolic geometry (or, in that sense, in geometry at all) I struggle ...
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64 views

Right angles in hyperbolic pool

(This uses a bit of physics) So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their ...
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393 views

Hyperbolic angle

I ve been looking in wikipedia and other sites for "hyperbolic angle", but it is not drawn anywhere. Only an area is shaded everywhere. Is it even possible to draw it?
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282 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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163 views

Structure of $x^2 + xy + y^2 = z^2$ integer quadratic form

The pythagorean triples $x^2 + y^2 = z^2$ can be solved in integers using rational parameterization of solutions to $x^2 + y^2 = 1$. It goes through $(1,0)$, then consider the line $y = -k (x - 1)$ ...
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408 views

Triangle inequality for hyperbolic distance

A quick way to define the hyperbolic metric in the Poincare disc is via the cross ratio: Given points a,b in the disc, let p,q be the endpoints of the hyperbolic line (halfcircle/line perpendicular to ...
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1answer
334 views

Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
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49 views

Consequence of the compactness of a hyperbolic surface

As part of a course I'm taking this semester, I am studying surfaces from this book http://www.math.brown.edu/~res/Papers/surfacebook.pdf. On page 142, the author presents a proof of the fact that ...
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125 views

Cusps for higher dimensional hyperbolic spaces

Take your favourite model of the hyperbolic plane $\mathfrak h$. It is well known how to define cusps, and we know those are $\mathbb Q \cup \{ \infty \}$. Now pick a higher dimensional hyperbolic ...
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1answer
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L is a family of hyp-lines passing through a pt. Not sure how this implies rr'=(c−Re(p))c'

Lemma 1. Let $p \in \mathbb{H}$, and assume $l$ is a family of hyp-lines passing through $p$ such that $l$ is of the form $l = \{c +re^{i\theta} | 0 < θ < π\}$. For simplicity, assume the ...
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1answer
120 views

centralizers in hyperbolic manifolds are cyclic

I am having trouble seeing why this statement is true: "If S admits a hyperbolic metric, then the centralizer of any non-trivial element of $\pi(S)$ is cyclic. In particular, $\pi(S)$ has trivial ...
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107 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb H3$, has a boundary $\mathbb CP1$. A ideal tetrahedron in $\mathbb H3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb CP1$. The four vertices ...
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486 views

Find the eclipse focal point

A conic with equation $$ a x^2 + b y^2 = c $$ has two focus points, where $a=4$, $b=24$ and $c=65$. One of those focus points has a positive x-coordinate. To two decimal places, what is the ...
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58 views

Polynomial behavior on hyperbolic plane

More a reference request / more information. I was reading some websites about hyperbolic geometry and got to thinking about how would polynomials $(x^2-2)$ behave in such a geometry. So, I need ...
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429 views

Lines in coordinate system of Hyperbolic Plane

An orthogonal coordinate system of the hyperbolic plain can be set up by fixing an orgio $O$, an $x$-axis, a $y$-axis (intersecting each other at $O$ in angle $90^\circ$), and, from any point $P$ ...
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1answer
62 views

hyperbolic trigonometric relation

Let $F$ be a hyperbolic once-punctured torus, and $G=\pi_1(F)$. Fix a discrete, faithful representation $\rho\colon G\to\mathbb{P}SL(2,\mathbb{R})$ and an element $g\in G$ corresponding to a ...