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

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1answer
108 views

How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
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2answers
175 views

convex polygons in hyperbolic geometry

Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.
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2answers
35 views

Hyperbolic quadrilaterals with two adjacent right angles

For convenience we'll work in the hyperbolic upper half plane $H$. We are given a hyperbolic quadrilateral $Q$ with vertices $a,b,c,d$ and geodesic segment edges $[ a,b ]$ $[ b,c ]$ $[ c,d ]$ $[ d,a ...
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0answers
220 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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1answer
61 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
138 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
84 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 ...
7
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2answers
246 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 ...
3
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1answer
90 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
231 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 ...
3
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2answers
146 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 ...
5
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1answer
94 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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1answer
89 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 ...
4
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1answer
85 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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1answer
143 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
51 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 ...
3
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1answer
197 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 ...
3
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2answers
229 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
212 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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1answer
348 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 ...
3
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1answer
275 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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0answers
30 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
52 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 ...
3
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1answer
184 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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0answers
151 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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1answer
128 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 ...
3
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2answers
655 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 ...
3
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2answers
707 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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0answers
101 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
84 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
82 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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0answers
403 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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0answers
38 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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1answer
52 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 ...
3
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0answers
56 views

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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2answers
94 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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6answers
617 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
126 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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1answer
52 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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2answers
358 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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0answers
212 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 ...
4
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2answers
158 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 $y = -k (x - 1)$ ...
4
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1answer
321 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
277 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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1answer
44 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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1answer
106 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
33 views

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 ...
4
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1answer
112 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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1answer
93 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 ...
3
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1answer
394 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 ...