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Questions tagged [hyperbolic-geometry]

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

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How can I prove that a systole of a close oriented hyperbolic surface S is a simple closed curve?

I looked into all the literature regarding systoles (ex. Katz) and everywhere I only see the statement, but without a proof. Also, if possible can anyone recommend me some articles or books on this ...
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18 views

Hyperbolic plane via parametrisation of the pseudosphere

My lecture notes (http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf), introduces the hyperbolic plane via a fundamental form on the pseudosphere that induces a metric on the ...
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25 views

Distance in upper plane doesn't depend on radius

Why when computing distance in upper plane, let say two points on a circle (denote them as x,y). The radius of the circle does not impact the distance, only the angle between them. Computing the ...
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17 views

Order in quaternion algebra for Fuchsian group

It may be that I am missing something very simple. In S. Katok's book "Fuchsian Groups", Lemma 5.3.3, we have the following. Lemma: Let $\Gamma$ be a Fuchsian group of finite covolume, $k_0=\mathbb{Q}...
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30 views

In the Upper Half-Plane, use the Triangle Inequality to show the shortest path from a point to a “vertical” line is an arc of a half-circle [on hold]

Consider the hyperbolic line $L=\{(0,y)\mid y>0\}$. Let $p=(a,b)\in \mathbb{H}^2$ be such that $a \neq 0$. By applying the Triangle Inequality in the triangle formed by the points $(a,b)$, $(−a,b)$...
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1answer
37 views

Given a hyperbolic triangle's sides (or angles), is there an easy way to determine whether it is inscribed in a circle, horocycle, or hypercycle?

If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
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22 views

If $P$ is a real quadratic polynomial, then $|P(z)| \geq \sqrt{\Delta} |Im(z)|$

I was reading an article which contains a statement about the distance in the hyperbolic plane: if $\gamma \in SL_2(\mathbb R)$ is hyperbolic then $\inf_{z \in H} d(\gamma z, z) = arcosh(Tr(\gamma)^2/...
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46 views

Word length vs hyperbolic length of curves on a hyperbolic surface

Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary. Fix some ...
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1answer
37 views

Scalar multiplication and vector addition with hyperbolic vectors

I'm representing hyperbolic vectors using the Minowski hyperboloid: $x_0^2 - x_1^2 - x_2^2 = 1$. I understand that the distance between two hyperbolic vectors of this form is $acosh(B(u, v))$ where $...
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Hyperbolic Geometry (circle)

Consider in $H^2$ the hyperbolic circle centered at $a + ib$ with radius $r$; i.e., the set $C = \left\{z\in H^2\mid dH^2 (z, a + ib) = r\right\}$ Show that $C$ is the Euclidean circle with center $a ...
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1answer
27 views

Poincaré disk construction

I am trying to understand how the Poincaré disk is constructed using the stereographic projection for the hyperboloid $x^2+y^2-z^2=-1$. So I want to project a line from the fixed point $(0,0,-1)$ to ...
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28 views

Calculating the distance function on a manifold, given the Riemannian metric in matrix form

I come from a CS background (and this is for a CS project) and as such my skills and knowledge of geometry are pretty poor. I'm looking at a bunch of points on a Riemannian manifold (let's say, for ...
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2answers
57 views

How does $f:z\mapsto az+b$ send circles in $\mathbb{C}$ to circles in $\mathbb{C}$?

I am looking at the proof in the beginning of chapter 2 of Anderson's book Hyperbolic Geometry, however I don't understand it. In particular, he seems to assume that $\overline{z_1} \cdot \overline{...
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1answer
76 views

Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$?

Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$? While evaluation a question on multiple integral I have got answer $4\sinh(3) \sinh(1)$. It was a multiple choice questions with a) $4\sinh(3) \sinh(1)$ ...
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1answer
38 views

Proving a Cauchy-Schwarz-like inequality

For real numbers $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ with $$x_1,y_1>0,\ x_1^2>x_2^2+\ldots+x_n^2,\ y_1^2>y_2^2+\ldots+y_n^2,$$ show that $$x_1y_1-x_2y_2-\ldots-x_ny_n \geq \sqrt{(x_1^2-...
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3answers
102 views

Circle whose radius is infinite

I have the intuition that a circle whose radius is infinite is a straight line. Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision. ...
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1answer
59 views

Number of points on a hyperbolic sphere a certian distance apart

Let $\mathbb{H}$ denote the upper half-plane with hyperbolic metric. Choose $p \in \mathbb{H}$. Let $\delta:=\lim\limits_{R\to \infty}\frac{\log(Vol\left(B(p,R)\right))}{R}$ be the volume entropy. ...
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1answer
26 views

Identifying the space of geodesics on the hyperbolic plane as a topological space.

On pg. 2 of this PDF, the author defines $\mathcal G=(-\infty, 0)\times(0, 1)$ and mentions that $\mathcal G$ can be thought of as "a space of geodesics on the hyperbolic $2$-space $\mathbb H^2$." ...
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1answer
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Is there a natural family of finite volume hyperbolic $3$-manifolds parametrized by $n$ distinct hyperbolic points?

Let $x_1,\ldots,x_n$ denote $n$ distinct points in the open $3$-ball, thought of as the Poincaré model of hyperbolic $3$-space. Let $X$ denote the complement of these $n$ points $x_i$, $1 \leq i \leq ...
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1answer
29 views

intersection of hyperbolic space and a cone in Lorentz space

As it is well-known, the geodesic sphere in the n-dimensional sphere $\mathbb{S}^n$ can be regarded as the intersection of $\mathbb{S}^n$ centered at the origin and a circular cone $$C=\{x\in\mathbb{R}...
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1answer
50 views

What means ''direction'' in hyperbolic geometry?

We can define the concept of ''direction'' as the equivalence class of parallel lines, but this is a good definition only if the Euclide parallels axiom is assumed, so that there is only one ...
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Find the equation of the hyperbolic line through $A=(3,4)$, perpendicular to hyperbolic line $x^2+y^2=25$, $ y>0$

I don't know how to solve this exercise in Hyperbolic Geometry. Find the linear equation of hyperbolic line which passes through point $A=(3,4)$ and it is perpendicular to hyperbolic line with ...
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36 views

Topics in hyperbolic geometry suitable for PhD-level research papers? [closed]

I am going to take admission in PhD course and I am planing to research on Hyperbolic Geometry. I saw some research on "hyperbolic distance or metric along the line of symmetry in a rectangle". ...
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35 views

Find the area of the hyperbolic (geodesic) triangle with vertices (0, 0),(0, 1) and (1, 0).

I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = \pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go ...
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1answer
26 views

Calculating side in quadrilateral in Poincare disc

If I have a quadrilateral ABCD inside the Poincaré disc such that $\angle A=\angle B=\frac{2\pi}{3}$, $AD=BC$ and we know the hyperbolic lengths of sides $AB$ and $CD$, how can I calculate the ...
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0answers
35 views

Equidistant points in hyperbolic space

I am unsure whether or not my proof for an exercise regarding equidistant points is correct. Let l be a hyperbolic line $\delta \gt 0$ and \begin{align*} E= & \; \{p \in H^2 \mid d(p,l)=\...
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0answers
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A geometric proof of Picard's little theorem

I'm preparing a presentation where I'd like to present a proof of Picard's little theorem using hyperbolic geometry. Picard's little theorem states that the range of an entire function can omit at ...
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1answer
29 views

Finding the norm of $w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $\mathbb{R}^n$

I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function $$M_w(z) = w + \frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z \in \mathbb{R}^n$ and $|w| < 1$. (...
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1answer
48 views

What does the following manifold look like?

In hyperbolic geometry, we know that any complete hyperbolic surface is the quotient $\mathbb{H}^{2}/\Gamma$ where $\Gamma < \text{Isom}(\mathbb{H}^{2})$ a discrete subgroup. We know $\text{Isom}^...
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15 views

Parallels and meridians on a pseudosphere (tractricoid)

I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere. On a sphere of radius $r$, using the usual $\theta, ...
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2answers
66 views

An identity for the Lorentz cross product

I need some help finding the error in my proof for an identity for the Lorentz cross product. For $x=(x_1,x_2,x_3)^T, y=(y_1,y_2,y_3)^T \ \in R^{2,1}$ the Lorentz cross product is defined as det[x,...
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41 views

Local Gauss-Bonnet for a rectangle

I'm trying to satisfy the local Gauss-Bonnet equation for the surface $R$ bounded by $u=A,u=B,v=a,v=b$ in the hyperbolic plane. However I am stuck. I know that the equation of local Gauss-Bonnet is $...
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67 views

Constructing right-angled regular polygons in the hyperbolic plane

I want to prove the existence of right-angled regular polygons in the hyperbolic plane. A polygon is regular if all its sidelengths and internal angles are equal. For which n exist right-angled ...
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Inversion as hyperbolic isometry (Poincaré disk model).

I'm beginning to study hyperbolic geometry in the Poincaré disk model, which is described as $$D = \{z \in \mathbb{C} : |z|<1 \},$$ and I need to show that inversion about a circle orthogonal to $\...
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1answer
104 views

Hyperbolic Geometry: Representation of the metric in a chart

Let $\mathbb{R}^{1,n}$ denote Lorentzian $n$-space, i.e., $\mathbb{R}^n$ equipped with the Lorentzian inner product $$\langle x,y \rangle = - x_0 y_0 + x_1 y_1 + \cdots + x_n y_n.$$ Define $\mathbb{H}...
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1answer
52 views

Picture of the Universal Cover of the Hyperbolic Pair of Pants

Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you ...
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32 views

Computing the length of a hyperbolic circle

I have a problem with the proof of the following exercise. Define the circle of radius $r \gt 0$ around a point c $ \in $ $H^2$ as $C_r(c)= \; \{x \in H^2 \mid d(c,x)=r\}$ where d($\cdot$,$...
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0answers
53 views

Proving a version of the Ultraparallel theorem

I have been asked to prove a version of the Ultraparallel theorem. Let $l_1$ and $l_2$ be hyperbolic lines with normals $n_1$ and $n_2$. Show that there exists a unique hyperbolic line $l_3$ ...
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1answer
52 views

Surjective differentiable map is an isometry

This is exercise 1.2 in Svetlana Katok's Fuchsian Groups. $\mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:\mathbb{H}\rightarrow\mathbb{H}$ is a surjective $C^1$ map. I want ...
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1answer
33 views

Perpendicular bisectors of hyperbolic lines

I want to prove the following basic property of hyperbolic lines in $IR^{2,1}$. If x $\in$ $H^2$ and l is a line in $H^2$ then there is a unique line l' through x orthogonal to l. I want to ...
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1answer
35 views

Why is the horizontal line $t\mapsto(t,y_0)$ not a geodesic in the Poincaré halfplane?

Let $\mathbb H=\{(x,y)\in\mathbb R^2\,|\,y>0\}$ be the Poincaré halfplane with metric $g=\frac{1}{y^2}g_\text{eucl}$ for the standard euclidean metric $g_\text{eucl}$. I know that the horizontal ...
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1answer
49 views

Calculating hyperbolic distance on the hyperboloid $x_1^2+x_2^2-x_3^2=-10000$

I want to calculate the hyperbolic distance on a hyperbolid defined by $x_1^2+x_2^2-x_3^2=-10000$. From wikipedia and other sources I got the formula that the distance for two points $x$ and $y$ on ...
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1answer
26 views

Iteration of a parabolic transformation

Let $a$ be a point of $S^{n-1}$ fixed by a parabolic transformation $\phi$ of $B^{n}$ (conformal ball model). One has to show that if $x$ is in $\bar{B^{n}}$ , then $$lim_{m \rightarrow \infty} \phi^...
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1answer
51 views

Finding the list of all closed curves in a “pair of pants” surface, having exactly 11 self-intersection points

Given a (small) natural number, for example $k=11$, is there any way to find all closed curves in an (arbitrary) "pair of pants" surface, having exactly $k$ (topologically non-removable) self-...
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2answers
181 views

If from (1, $\alpha$) two tangents are drawn on exactly one branch of the hyperbola $\frac{x^2}{4} -\frac{y^2}{1} = 1$ the alpha belongs to [closed]

If from (1, $\alpha$) two tangents are drawn on exactly one branch of the hyperbola $$\frac{x^2}{4} -\frac{y^2}{1} = 1$$ the alpha belongs to As far as I can see two tangents can be drawn to ...
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0answers
17 views

Hyperbolic Quadrilateral

Suppose we are given four angles $\alpha$, $\beta$, $\gamma$ and $\delta$. For a quadrilateral to exist with interior angles $\alpha$, $\beta$, $\gamma$ and $\delta$ in the hyperbolic plane we must ...
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0answers
19 views

Proof of lemma 2.4 in Cantat-Lamy's paper

I am currently reading the paper 'Normal subgroups in the Cremona group'. I am stuck at the proof of lemma 2.4 where the author claimed that 'One easily checks that $p_3$ is contained in $[p_1,p_2]$ ...
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1answer
64 views

What are the transformations that preserve cross ratios on a sphere in higher dimensions?

If we have four points $x,y,z,w$ on a sphere, then the cross ratio is $\frac{|x-z|}{|x-w|}\frac{|y-w|}{|y-z|}$. If we consider $S^1 \subseteq \mathbb{C}$, then the transformations of $\mathbb{C}$ ...
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1answer
33 views

Find explicitely an element of SL$_2(\mathbb Z)$

Let $\Gamma=\text{SL}_2(\mathbb Z)$ act on $\mathbb H=\{z \in \mathbb C: Im(z) > 0\}$ by $\gamma z=\frac{az+b}{cz+d}$, where $\gamma$ is the matrix $\left[\begin{matrix} a & b \\ c & d \end{...
3
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1answer
36 views

The collapsing map and its coarse inverse are $32 \delta$-coarse Lipschitz to each other

TL,DR: Why do we have $$d(\overline{\kappa} \circ \kappa, Id) \leq 32 \delta?$$ I am reading Chapter 11 from the book "Geometric group theory" by Cornelia Druţu and Michael Kapovich (freely available ...