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Questions tagged [hyperbolic-geometry]

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

4
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1answer
34 views

Universal cover of a not-necessarily-complete hyperbolic manifold

The Cartan-Hadamard theorem (as typically stated) tells us that the universal cover of a geodesically complete and connected Riemannian manifold $M$ with non-positive sectional curvature is ...
-1
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1answer
48 views

Simplification of $\sin(\pi^x)$ , with $x$ being a positive irrational number

How to simplify if $a > 0$ and $\cos(a) < 0$ Was a previous post. Correction, it was suppose to be if a > 0 & cos(a) > 0. An answer was given. https://math.stackexchange.com/a/1274372/...
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0answers
13 views

non-uniformly hyperbolic definition

I'm working on Uniformly hyperbolic finite-valued $SL(2,R)$ -cocycles( article from Arthur Aveila and Jairo Bochi)and at the beginning of my researches,i want to know the exactly meaning of the title ...
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0answers
17 views

Hyperbolic isometry and line segments

I was trying to apply Poincare's Polygon theorem, for that I had to give a pairing of sides, i.e., to have an isometry of the hyperbolic plane that will take a side of a polygon to another side (of ...
0
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1answer
20 views

How to prove the existence of infinite geodesics that do not intersect in hyperbolic space

My given question is: In hyperbolic space, given a geodesic $L$ and a point $p$ not lying on $L$, show there is an infinite number of geodesics through $p$ which do not intersect $L$. The "model" ...
16
votes
2answers
707 views

Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such ...
0
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1answer
13 views

Right triangle circumscribed by a horocycle

Is it possible that a right triangle is circumscribed by a horocycle? Or, is this statement a theorem in the hyperbolic gemetry? For any horocycle $\gamma$, there are no three distinct ordinary ...
1
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0answers
21 views

Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
0
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1answer
43 views

Can this equation $b^2$ = $c^2-a^2$ be derived intuitively?

Today while proving the equation of hyperbolas,$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\tag{1}$$ I came across this expression $$b^2=c^2-a^2\tag 2$$ Though this expression seems much like Pythagorean ...
1
vote
1answer
39 views

Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
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0answers
25 views

Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
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0answers
20 views

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
0
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1answer
43 views

Prove that hyperbolic isometry with constant distance is the identity

Let $f$ be an isometry of the Poincaré half-plane model of two-dimensional hyperbolic geometry, denoted by $\mathbb{H}^2$. Prove that if the distance $d(z,f(z))=c$ for some constant $c\geq0$ for all $...
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0answers
7 views

boundary of tubular neighborhood of convex subset with piecewise smooth boundary

The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is known to be smooth, e.g., by page 73 of Hyperbolic Manifolds and Kleinian Groups. I wonder whether this is true for ...
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0answers
23 views

Proof bijectivity of map between a hyperbolic plane $\mathcal{H}^2$ and open unit disk in $\mathbb{R}^2$

Given a hyperbolic plane $\mathcal{H}^2=\{(t,x,y)\in\mathbb{R}^3|t^2=x^2+y^2+1, t>0\}$, I had to construct a function $f$ that maps as follows: For a point $P$ in $\mathcal{H}^2$, draw a line $l$ ...
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0answers
20 views

Ideal tetrahedron maximum volume

I want to that any tetrahedron in $\Bbb{H}^3$ can be transformed into a tetrahedron that has $0, 1, \infty, z$ as vertices. Also I need to show that the above defined tetrahedron has maximum volume if ...
0
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0answers
11 views

Examples of isometry group

I would like to find some isometries groups G of a $\delta$-hyperbolic spaces such that $ \rho(G)=inf_{g \neq e}(||g||)>4\delta$. I already find $\mathbb{Z}$ acting on $\mathbb{R}$ that verify that....
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0answers
38 views

Find locus of points of constant distance from the imaginary axis in the Poincaré upper-half plane model

I have been asked to find the locus of points of constant distance $d$ from the vertical axis $\{ \textrm{Re}(z)=0\}$ in the Poincaré half-plane model in hyperbolic geometry. Here are my thoughts on ...
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0answers
44 views

Random geodesic on Bolza surface

On a standard flat torus, we can construct a random geodesic (i.e. a dense geodesic on torus) by choosing at the beginning an irrational direction from the point (0,0). My question is how can we ...
4
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0answers
31 views

Valid metric on a hyperbolic space

Note: cross-posted to mathoverflow.net I'm looking at the distance that's defined in this paper on Poincaré Embeddings: $d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...
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0answers
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Complex-analytic isometry of hyperbolic ball and half-space

I'm trying to prove that the $n$-dimensional Poincaré ball and half-space models of hyperbolic space are isometric. Here the Poincaré ball $\mathbb B^n_R$ ($n$-ball of radius $R$) has the Riemannian ...
1
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1answer
23 views

Riemann Mapping theorem in triangulations

I am reading the paper 'Rotation Distance, Triangulations, and Hyperbolic Geometry' by Thurston et al. The authors are constructing a sequence of triangulation from a regular icosahedron. Each face of ...
4
votes
1answer
25 views

Poincaré disk with hyperbolic metric is a metric space

I am trying to prove that the Poincaré disk $D=\left\{ z \in \mathbb{C} : |z|<1 \right\}$ equipped with the hyperbolic metric given by $d_{D}(z_1,z_2)=\inf \{ L_{D}(\gamma) \mid \gamma \text { is a ...
1
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1answer
28 views

What is the implied metric on $SL(2,\mathbb{R})$ in the definition of Fuchsian group?

I'm studying Fuchsian groups, which are by definition discrete subgroup of $SL(2,\mathbb{R})$. Authors often do not tell, though, what topology (metrizable I assume) is taken on $SL(2,\mathbb{R})$ and ...
5
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1answer
58 views

Hyperbolization Theorem

I know that for experts this question is trivial, but it's been a while I'm having trouble understanding this... The version of the hyperbolization theorem I found on Aschenbrenner, Friedl and Wilton'...
2
votes
1answer
39 views

Rectangles in Hyperbolic geometry

This says rectangles don't exist in hyperbolic geometry. But according to this you can model the euclidean plane in Hyperbolic geometry. Wondering why you can't model the rectangle, I thought if you ...
2
votes
1answer
19 views

Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$

Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
2
votes
1answer
47 views

Finding the point through which the tangent passes

A hyperbola passes through the point $P=(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0)$. Show that the tangent at $P$ passes through the point $(2\sqrt{2},3\sqrt{3})$. Attempt: So the equation of ...
0
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1answer
22 views

geodesics in hyperbolic space

Let $M$ be the Poincare ball model of the Hyperbolic space, and let $\zeta \in T_0M$. In my lecture notes it is claimed that $$c(t)=\tanh(\Vert \zeta \Vert t )\zeta/\Vert \zeta \Vert$$ is the geodesic ...
0
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1answer
18 views

a good description of the Cayley--Klein models especially about its homogeneity property

Actuality, I'm working with conic in hyperbolic geometry and I'M looking for a good description of the Cayley--Klein models especially about its homogeneity property?
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2answers
61 views

Geodesics of Poincaré Ball Model

I am asked to consider the Poincaré ball model of 3d hyperbolic space and characterize its geodesics. I.e. we have a pseudo-riemannan manifold $(M,g)$ where $$ M = \{(x^1,x^2,x^3)\in \mathbb{R}^3 : \|...
0
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0answers
23 views

Hyperbolic circles, horocycles and hypercycles

How do you write a formula for Hyperbolic circles, horocycles and hypercycles in Normal form give a point of rotation, or translation? Specifically, a hyperbolic rotation (hyperbolic circle) about ...
0
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1answer
22 views

Question regarding the kernel of $\Lambda:\operatorname{SL}(2)\rightarrow \operatorname{Isom}(\Bbb H_2)$

Let $\Lambda:\operatorname{SL}(2)\rightarrow \operatorname{Isom}(\Bbb H_2)$ be defined by $\Lambda(g)(z)=\frac{g_{(1,1)}z+g_{(1,2)}}{g_{(2,1)}z+g_{(2,2)}},g\in SL(2),z\in \Bbb H_2$. I need to check ...
0
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1answer
16 views

What is the group operation of $Isom(\Bbb H_2)$?

I have been given the task of showing that the function $\Lambda:SL(2)\rightarrow Isom(\Bbb H_2)$ defined by $\Lambda(g)(z)=\frac{g_{(1,1)}z+g_{(1,2)}}{g_{(2,1)}z+g_{(2,2)}},g\in SL(2),z\in \Bbb H_2$ ...
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0answers
24 views

How to sample from a multivariate normal distribution in hyperbolic space?

I want to generate samples from a multivariate normal distribution on the hyperbolic space, such that $\forall x \in \mathbb{H},$ $$ \text{pdf}(x) = \frac{1}{c} e^{-\frac{d^2_{\mathbb{H}}(x,\mu)}{2\...
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0answers
46 views

Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
0
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1answer
52 views

Description of Model of Euclidean Geometry found in the Hyperbolic Plane

I have read that there is a model of Euclidean Geometry in the Hyperbolic Plane, but can't find any description on the web in a digestible form and thought I'd ask this question: If one can describe ...
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0answers
16 views

Loxodromic element of an hyperbolic group

I struggle to show the following thing : Let $G$ be an $\delta$-hyperbolic group related to the space X, $g$ be a loxodromic element. Then $E(g)=\lbrace u \in G | \exists \in \mathbb{N}, ~ ug^ku^{-1}=...
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1answer
43 views

Find the hyperbolic distance between $2$ and $5+i$ in the upper half plane $H=\{ z: Im(z)>0\}$.

Find the hyperbolic distance between $2$ and $5+i$ in the upper half plane $H=\{ z: Im(z)>0\}$. Ans: we know the metric $d_H(z, w)=2\tanh^{-1}(|\frac{z-w}{z-\bar w}|)$ then $d_H(2, 5+i)=2\tanh^{-...
1
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1answer
22 views

Inner product on the Hyperbolic half plane

I think this is the dual question to my previous question. From Gudmundsoon notes, page 60: We can model the hyperbolic space $\mathbb H^m$ as the super half plane space $\mathbb R^+ \times \mathbb ...
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1answer
32 views

Isometries of a surface with metric of curvature $-1$

Let $\mathbb{H}^2$ be the hyperbolic space in the model $$\mathbb{H}^2=(\mathbb{R}\times\mathbb{R}_+,g=\frac{1}{y^2}(dx^2+dy^2)).$$ It is known that the Mobius transformations, with $ad-bc=1$, are ...
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1answer
22 views

Surface covered by hyperbolic plane admits a hyperbolic metric

Let $S$ be a surface. Is it true that if $S$ is covered by the hyperbolic plane (or a subset thereof) then it admits a Riemannian metric of constant negative curvature? How does the metric (or ...
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0answers
49 views

Hyperbolic metric and its relation to the distance between matrices

There is a bijection $f$ between $PSL(2,\mathbb{R})$ and $T^1\mathbb{H}$, which sends a matrix $M$ to a vector with base point $\frac{M_{11}i+M_{12}}{M_{21}i+M_{22}}$. In particular, the identity ...
3
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0answers
41 views

Question about circles in hyperbolic space

I'm trying to prove the following: In the half-plane model of $\mathbb H^3$ let $L$ be the geodesic going from the origin of $\Bbb R^3$ to $\infty$. So $L$ is a straight half-line perpendicular to $\...
2
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1answer
40 views

derivative action of isometries on hyperbolic 3-space (upper half-space model)

Let $\mathcal{H}^2=\{z=x+iy\in\mathbb{C} : y>0\}$ be the upper half-plane and let $g(z)=\frac{az+b}{cz+d}$, $a,b,c,d\in\mathbb{R}$, $ad-bc=1$, be an orientation preserving isometry of $\mathcal{H}^...
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2answers
109 views

Is there a theory of “hybrid” geometries?

Standard 2D geometries, elliptic, Euclidean and hyperbolic, can be all derived from the same basic idea: start with projective geometry formed by lines and planes through origin in $R^3$ and then put ...
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1answer
80 views

Group generated by all inversions in hyperbolic lines

In Groups and Geometry by Lyndon, Chapter 9 part 3 (page 165) started by introducing a new group (denoted as $\widetilde{H}$ below) and a theorem. Let $H$ denote the hyperbolic group and $H^{+} = \{ ...
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0answers
36 views

Troublesome theorem in hyperbolic geometry

I want to prove the following theorem without using Dedekind's axiom(i.e. only with axioms of hyperbolic plane) Given arbitrary line $\mathcal l$ and a point $\mathcal P$ which does not lie on $\...
2
votes
2answers
169 views

What is $T^1(\mathbb H^2/PSL_2(\mathbb Z))$?

Let $\mathbb H^2$ be the upper-half plane. The group $PSL_2(Z)$ acts on $\mathbb H^2$ by isometries, and hence we get an action on $T^1(\mathbb H^2)$. This action is free, smooth, and proper, and thus ...
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0answers
29 views

How to compute expansion factors for hyperbolic rational maps?

It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...