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

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4
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1answer
270 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 ...
0
votes
1answer
234 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 ...
0
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1answer
43 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 ...
1
vote
1answer
96 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 ...
1
vote
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
votes
1answer
105 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 ...
1
vote
1answer
80 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
votes
1answer
329 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 ...
0
votes
1answer
55 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 ...
2
votes
0answers
298 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 $P$ ...
2
votes
1answer
56 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 ...
1
vote
1answer
190 views

Length and area in hyperbolic geometry

I am reading a book about modern geometries by Michael Henle. He gives formulas for length of a curve and area of a region (in upper half plane model: $l(\gamma)=\int _a^b \frac{|z'(t)|}{y(t)}dt, ...
2
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1answer
56 views

Character Varieties- reference request

I will start learning about character varieties. I need to learn about Teichmuller spaces and how to consider them as components of "some character variety". Can someone recommend some textbooks or ...
3
votes
1answer
155 views

Volume of a Riemannian manifold and its relation to the area

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
3
votes
0answers
141 views

How do we define a complete metric on a Riemann surface with punctures?

This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this ...
2
votes
1answer
108 views

Lift of a homeomorphism $f$ between two (hyperbolic) surfaces $X,Y$

Let $X,Y$ be two hyperbolic Riemann surfaces (i.e. they have universal cover the upper half plane $\mathbb{H}$). Let $\pi_X:\mathbb{H}\to X, \pi_Y:\mathbb{H}\to Y $ be the corresponding covering maps. ...
3
votes
1answer
275 views

Riemann surface with punctures corresponds to a hyperbolic surface with cusps

I am reading a paper on Riemann surfaces and the author used the fact that $\{$Riemann surfaces with genus $g$ and $n$ punctures$\}$ is in one-to-one correspondence with $\{$ hyperbolic surfaces ...
5
votes
1answer
214 views

Parabolic elements correspond to punctures

In Mapping Class Group by Farb and Margalit page 22, they say: Let $S$ be a hyperbolic surface. If a non-trivial element of $\pi_1(S)$ is represented by a loop (up to homotopy) around a puncture, ...
4
votes
1answer
240 views

Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?

There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations ...
6
votes
1answer
226 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
1
vote
1answer
126 views

fixed point of a parabolic commutator

For a hyperbolic isometry $g$ of the hyperbolic plane, denote by $ax(g)$ the oriented geodesic running from the repelling fixed point $p_g$ and the attracting one $q_g$. Let $G$ be the fundamental ...
3
votes
4answers
2k views

Distance in the Poincare Disk model of hyperbolic geometry

I am trying to understand the Poincare Disk model of a hyperbolic geometry and how to measure distances. I found the equation for the distance between two points on the disk as: $d^2 = (dx^2 + dy^2) ...
2
votes
1answer
109 views

radii of horoballs

Consider the horoball $$B_h=\left\{(z,t)\in\mathbb{H}^3\mid t>h>0\right\}.$$ If $T\in\mathbb{P}SL(2,\mathbb{C})$ is an isometry of the hyperbolic space which does not fix the point at infinity, ...
5
votes
1answer
145 views

Conformal automorphism of $H^n$

I was looking for the characterization ( or a complete list ) of the conformal automorphisms of the upper half space $H^n$ in $R^n$. I know that when $n=2$, it is $PSL(2,R)$ and when $n=3$, it is ...
4
votes
1answer
57 views

Can different uniformizations of Riemann surfaces be related somehow

Let $X$ be a hyperbolic compact connected Riemann surface. Let $U\subset X$ be an open subset. Assume that $U\neq X$. We can uniformize $X$ by $\mathbf{H}$ directly to obtain it as a quotient of ...
5
votes
3answers
825 views

Expression of the Hyperbolic Distance in the Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. ...
5
votes
2answers
195 views

Reflections generating isometry group

I was reading an article and it states that every isometry of the upper half plane model of the hyperbolic plane is a composition of reflections in hyperbolic lines, but does not seem to explain why ...
2
votes
2answers
90 views

Solving for the Moebius transform that sends $(x_0, y_0)$ to $(x_1, y_1)$ with two known fixed points

Suppose $M$ is a hyperbolic Moebius transformation with fixed points at $(0, 0), (1, 0)$ which, when applied to the complex $(x_0, y_0)$, yields the result $(x_1, y_1)$. How do I solve for $M$ given ...
1
vote
1answer
76 views

Limit sets of representations of once-punctured torus groups and circle packings

Let $\rho\colon\pi_1(T_1)\to PSL(2,\mathbb{C})$ be a faithful representation of the fundamental group of a once-punctured torus. If both the components of the convex core in the quotient manifold are ...
0
votes
1answer
315 views

Hyperbolic coordinates

You can uniquely specify any point in 2D Euclidian space using 2 numbers: the distance from the infinitely long X-axis, and the distance from the infinitely long Y-axis. How do you uniquely specify a ...
4
votes
1answer
120 views

Fuchsian groups and surfaces

It's a fact that if two Fuchsian group are conjugate, the corresponding surfaces are isometric. Is the converse true ? Take 2 isometric Riemann surfaces $S$ and $S'$(which are covered by the upper ...
2
votes
0answers
85 views

Distance realizing geodesic in a hyperbolic surface

Suppose $S$ is a hyperbolic surface with geodesic boundary and $P$ is a hyperbolic pair of pants with $a$, $b$, $c$ geodesic boundary. Let $\gamma$ be the unique geodesic realizing the distance ...
3
votes
2answers
473 views

How does a conformal mapping preserve angles in hyperbolic geometry?

Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = ...
0
votes
1answer
186 views

Geodesic distance on complex upper half plane

Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane. Does there exist an $\epsilon >0$ such that the geodesic distance ...
12
votes
3answers
2k views

how to generate tesselation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's ...
5
votes
1answer
500 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
1
vote
1answer
163 views

A book to study about hyperbolic plane, hyperbolic translations, etc.

In this paper, page $6$, the authors state the following: The translations of the hyperbolic plane are defined as products of two central symmetries; the set of hyperbolic translations forms a ...
6
votes
1answer
199 views

When does there exist an isometry that switches two subspaces?

Let $V$ be a real vector space of finite dimension and let $\langle \cdot, \cdot \rangle$ be a non-degenerate symmetric bilinear form on $V$. Let $U, W \subseteq V$ be linear subspaces such that ...
3
votes
1answer
295 views

Thrice-punctured sphere

This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45: the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric. I ...
4
votes
1answer
192 views

Hyperbolic triangle and two points in Poincare disk

Given a hyperbolic triangle $T$ and two points $p$ and $q$ in Poincare disk. Note that $p$ and $q$ are outside the triangle. If $p$ has shorter distances to the three vertices of $T$ than $q,$ can we ...
2
votes
1answer
130 views

Bounding the number of integer solutions of the following inequality

Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed. How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
0
votes
1answer
125 views

Geometry of a hyperbolic triangle

How can we find the upper bound of the hyperbolic distance from any point on the side $AB$ to either $AC$ and $BC$ for the hyperbolic triangle $\triangle ABC$? Help will be appreciated.
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0answers
172 views

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 ...
3
votes
1answer
181 views

Wikipedia article on Hyperbolic geometry

I was reading the Wikipedia article on hyperbolic geometry and have come across the line geodesic paths are described by intersections with planes through the origin Why is this necessarily ...
4
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1answer
175 views

Characterization of linearity in terms of metric

At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements '$y$ lies on the line segment determined by $x$ and $z$ ' and '$d(x,y)+d(y,z)=d(x,z) $' are ...
3
votes
1answer
152 views

Why are perpendicular bisectors 'lines'?

Given two points $p$ and $q$ their bisector is defined to be $l(p,q)=\{z:d(p,z)=d(q,z)\}$. Due to the construction in Euclidean geometry, we know that $l(p,q)$ is a line, that is, for $x,y,z\in ...
1
vote
1answer
194 views

Real part of $\sinh(z)$

Is it true that for every $z \in \mathbb{C}$ $$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z ...
3
votes
2answers
524 views

Does anyone know a good hyperbolic geometry software program?

We are currently using this program called NonEuclid but it is a little frustrating to use sometimes and I was wondering if anyone knows another program for hyperbolic geometry.
2
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0answers
235 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
4
votes
6answers
225 views

Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...