Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.

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6 views

how to find coordinates of a heptagons using the Poincare disk model? [closed]

I want to know how I can find cartesian coordinates of heptagons generated on a Poincare disk.
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Thurston Compactification

Sorry in advance for my English. I'm studying the Thurston compactification from the Jean-Pierre Otal's book "The Hyperbolization Theorem for Fibered 3-Manifolds". I have a question, what $\mathbb{...
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33 views

How to show that two arcs are parallel with respect to poincare metric of the unit disc?

Show that two circular arcs in the unit disc with common end points on that unit circle are noneuclidean parallels in the sense that the points on one arc are at constant distance from the other. For ...
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1answer
30 views

Hyperbolic Geometry - Triangle Bisectors Proof

I'm refreshing myself on hyperbolic geometry using Wolfe's "Introduction to Non-Euclidean Geometry". This is problem number 6 from page 81 of that text: "Show that a line through the midpoint of one ...
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1answer
109 views

Examples of non-positively Curvature Riemannian Manifolds

When I read about complete, simply connected, and connected Riemannian manifolds of non-positive curvature I only find explicit examples of hyperbolic $n$-space and Euclidean space. What are other ...
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1answer
21 views

Orbits of points under pseudo-Anosov diffeomorphisms

I have no intuition about orbits of pseudo-Anosov diffeomorphisms $\phi$ of closed surfaces $S$ of genus $>1.$ I understand that there are infinitely countably many periodic points, correct? What ...
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56 views

Extending a quasi-isometry of a neutered hyperbolic space

Suppose $\phi : B \to B$ is a quasi-isometry of a neutered space $B$ (so $B$ is obtained by removing a collection of disjoint open horoballs from $\mathbb{H}^n$, and the metric $d_B$ on $B$ is the ...
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1answer
33 views

Diameter of triangle in the hyperbolic plane

The diameter of a set $S$ in a metric space $(M,d)$ is defined to be {$\sup d(x,y)|x,y\in S$}. In Euclidean space the diameter of a triangle is the length of the largest side. In the hyperbolic ...
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27 views

Seeking a formal definition of a complex hyperbolic space

I'm trying to find a formal definition of a complex hyperbolic space that is $CH^n$. Can anyone help me please? I tried looking online already but no formal direct definition.
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How to prove uniqueness?

For the question asked in Is a Möbius transformation that PRESERVES UNIT DISC uniquely determined by three distinct points and their images? , how can I prove the UNIQUENESS of those ...
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Linear Fractional Transformations on Ideal Triangles [closed]

Assume an ideal triangle is given in a hyperbolic geometry, with points meeting the x-axis at $-a$, $0$, and $a$. All angles are equal to $0^o$. I'm trying to understand how one would find a linear ...
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How to pushforward a metric in special cases?

I was told by someone a sketch to prove little Picard theorem as follows. Step 1. The elliptic modular lambda function $\lambda$ gives a holomorphic covering map $\mathbb{H}\rightarrow \mathbb{C}-\{0,...
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2answers
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Solving $R\space \sinh\frac{D}{R}=k$ for $R$

Does a solution exist for $R$ in this equation? I can't seem to solve it either analytically or numerically. $$R\space \sinh\frac{D}{R}=k$$
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How to derive relations between the sides and angles of equilateral hyperbolic triangles

I hope everyone in this community is staying safe, well and isolated. In this unprecedented situation I am starting to learn about some non-Euclidean geometry and explore down a fractal. In the ...
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1answer
58 views

When might Fenchel-Nielsen twist coordinates exceed 1/4?

When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length ...
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How to find the dual point of a line geometrically?

I'm learning hyperbolic geometry by following Prof NJ Wildberger videos on YouTube. So far I know how to find the dual of a point (which is a line) geometrically (using pen, paper, and ruler). Here's ...
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Hyperbolic manifolds that are quotient of hyperbolic space

If $M=\mathbb{H}^n/\Gamma$ is a hyperbolic manifold ($\Gamma$ being a group of isometries of $\mathbb{H}^n$) can I conclude that it is complete? I know that if the covering map was a local isometry ...
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115 views

First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
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Surfaces of non-constant negative curvature

Are there any nice models or books/papers with elementary discussions of surfaces with non-constant negative curvature, but negative everywhere, analogous to the Poincare disk for constant negative ...
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1answer
45 views

Geometric proof that the area of a hyperbolic triangle is proportional to its angle defect

The area of a spherical triangle with angles $\alpha$, $\beta$ and $\gamma$ on the 2-dimensional unit sphere is $\alpha + \beta + \gamma - \pi$. There is a nice geometrical proof of this fact that ...
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Model for $2$-dimensional manifold with constant negative sectional curvature

Let $M$ be a simply connected, complete manifold of dimension $2$ and constant sectional curvature $-k$ for $k>0$. I know I can model this manifold taking the circle of radius $1/\sqrt{k}$ and ...
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Particular configuration of planes and lines in three dimensional hyperbolic geometry

This is a question about three dimensional hyperbolic geometry. I don't know if there is any notion for this but let us call three lines H-parallel whenever they are pairwise disjoint, contained in ...
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39 views

area in hyperbolic geometry

suppose p is smaller or equal to q, and r is strictly greater than 0 and smaller or equal to s, where p,q,r,s are constants. how do I find the hyperbolic area of the region [p,q]x[r,s] in the ...
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1answer
29 views

How to compute the side length of an equilateral hyperbolic triangle from its angle?

In the hyperbolic plane we can have equilateral triangles with any angle smaller than $\pi/3$. The angle $\alpha$ determines their full shape. The area is easily obtained from $\alpha$ as $\pi - 3\...
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23 views

length-difference in a hyperbolic geodesic triangle

Let $c_1:[0,t_1]\rightarrow \mathbb{H}^2$ and $c_2:[0,t_2]\rightarrow \mathbb{H}^2$ be two unit-speed geodesics in the hyperbolic plane with $c_1(0)=c_2(0)$ and such that the angle between $c_1'(0)$ ...
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If $R(x)=\inf\{R>0; K\subset\overline{B_{{\rm hyp}}(x,R)}\} $ there exists a unique $p\in\Bbb{H}$ such that $R(p)=\inf\{R(x);x\in\Bbb{H}\}$?

Considering the function $R(x)=\inf\{R>0; K\subset \overline{B_{{\rm hyp}}(x,R)}\} $, I need to prove that there exists a unique $p\in \mathbb{H}$ such that $R(p)=\inf\{R(x); x \in \mathbb{H}\}...
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27 views

For what reason straight lines must be on planes that go through the origin and how were the centers (origin) of the different geometries defined?

I've asked this to many mathematicians but I don't get a conclusive answer. Regarding origins (centers): - I understand that the origin in spherical geometry is the equidistant to all the points on ...
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1answer
42 views

Which automorphisms of the plane preserve the hyperbola?

Is there a reasonable characterization of all the "power-law" diffeomorphisms of finite order $f:\mathbb R^{>0} \times \mathbb R^{>0} \to \mathbb R^{>0} \times \mathbb R^{>0}$, of the form ...
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Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?

Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm? Specifically, consider the poincare half-plane model of the 2d hyperbolic ...
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What is the relation between De Sitter space and the Forward sheet of Hyperboloid Model?

The forward sheet of hyperboloid is given by, $-z_0^2 + z_1^2+ ... z_n^2 = -a^2$. The induced distance function is 1) $d(x,y) = arcosh(-\langle x,y \rangle)$. The de Sitter space is defined as - $-...
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57 views

Is Shadowing Lemma specific to hyperbolic dynamical systems?

In a hyperbolic dynamical system, the Shadowing lemma states that every epsilon-pseudo-orbit is uniquely delta-shadowed by some orbit. see: https://en.wikipedia.org/wiki/Shadowing_lemma It is not ...
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Poincare distance of a point from Euclidean circle

I’m considering Poincaré disk model. Let $z$ be any point in $D$, with $|z|>r$, where $0<r<1$. I need to find Poincare distance of this point from the circle $\{z:|z|=r\}.$ What I think is, ...
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Reparametrising a 1st Fundamental Form to the 1st F.F. of the half-plane model of the hyperbolic plane

I'm struggling to complete an exercise that gives a reparametrisaton to use to convert one 1st Fundamental form into that of the half-plane model of the hyperbolic plane. I've written out the exercise ...
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1answer
28 views

Parallellogram in Hyperbolic Geometry is not composed of two congruent triangles.

I am trying to show that in hyperbolic geometry, any diagonal of a parallellogram $\square ABCD$ divides the parallellogram into two non-congruent triangles. I have tried assuming the contrary, and ...
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1answer
23 views

Constructing a point on a geodesic line in the hyperbolic disc with a given distance from another point

Given a geodesic line $L$ in the Poincaré disc $\{ (x,y) \in \mathbb{R}^2 : |x|^2 + |y|^2 < 1 \}$ with the hyperbolic metric and a point $P \in L$, how can one construct a point $Q \in L$ that has ...
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How do determine if two matrices in the modular group are equivalent as elements of (2,3,7)

Following up from How do you obtain the $(2,3,7)$ triangle group as a quotient of the modular group?, I'm trying to understand the (2,3,7) triangle group as a quotient as the modular group. So, we ...
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25 views

Existence of compact space forms of negative curvature

Do there exist compact Riemannian manifolds of constant negative sectional curvature in all dimensions $\geq2$? The two-dimensional case is well-known. For higher dimensions, this question asks for ...
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1answer
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How do you obtain the $(2,3,7)$ triangle group as a quotient of the modular group?

So, at various places on Wikipedia (like here), its claimed that the $(2,3,7)$ triangle group can be obtained as quotient of the modular group. How is this done? Treating the integers as mod $7$ ...
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48 views

Heegaard genus and ranks of fundamental groups

I found an interesting conjecture in the paper "Hyperbolic volume, Heegaard genus and ranks of groups" by Peter B. Shalen. It says that if $M$ is a compact, orientable, hyperbolic 3-manifold, then ...
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Coordinate System for the Hyperboloid Model

I am trying to find a coordinate system for the hyperboloid model. The n-dimensional case. It is obviously something like the spherical coordinates in n-dimensions I just can't wrap my head around the ...
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1answer
58 views

Writing an algorithm solving the word-problem in hyperbolic groups

I am reading in the “Metric Spaces of Non-Positive Curvature Book by André Haefliger and Martin Bridson”, on Dehn's Algorithm (Chapter III.Γ, p.449). Let $\mathcal{A}$ be a finite generating set of ...
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1answer
28 views

How can I find a medium-length word equal to 1 on a 2-torus?

I'm trying to give an example of Dehn's algorithm on a 2-torus and to do so I want to find a word about 20 letters or so in length that is equal to 1 so that I can apply the algorithm. I'm having ...
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Logistic function symmetric (?) and $\tanh$ asymetric

From Modern Multivariate Statistical Techniques (Izenman), Exercise 10.2: "Show that the logistic function is symmetric, whereas the tanh function is asymetric." In the exercise immediately prior, ...
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1answer
34 views

What regular or semiregular hyperbolic tiling has the smallest average tile area?

I have noticed that hyperbolic tilings tend to be rather "sparse" in that each tile takes up a lot of space. If I remember correctly, for a given curvature the area of any tile in a given hyperbolic ...
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conversion of a metric from hyperbolic to Riemann normal coordinates

We consider a hyperbolic half-plane (also called Poincar ́e upper half plane), $$\mathbb{H}^{2}=\left\{ \left(x,y\right)\Biggl|y>0\right\}$$ . It has got a simple 2D Riemannian metric with open ...
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1answer
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Question on cusp link for a hyperbolic 3-manifold

I'm currently reading Ratcliffe's Foundations of Hyperbolic Geometry (third edition). I'm having trouble resolving a possible contradiction in one of the theorems. Let $M$ be a hyperbolic $3$-...
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56 views

Do there exist space-filling curves that fill the whole hyperbolic 2d plane? If so, can they be visualized?

Just as mentioned in a similar question, in Euclidean plane, there exist space-filling curves that fills the whole plane. So for hyperbolic plane, do there exist some space-filling curves?
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1answer
107 views

a long product of elements in hyperbolic group is not a proper power

Let $G$ be a hyperbolic group, i.e., there exist $\delta>0$ and a finite generating set $S$ of $G$ such that the Cayley graph $X$ of $G$ relative to $S$ is a $\delta$-hyperbolic space. Assume also ...
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2answers
31 views

Find tangents at a specified positions

I have two similar problems that I have tried to solve for several hours now but I end up with wrong answers. So there is something I do not understand correctly. Problem 1: The intersection between ...
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20 views

Formula for action of Möbius transformation on the hyperboloid model

The group of Möbius transformations are isomorphic to the group of orientation-preserving isometries of hyperbolic space. The 3-dimensional hyperboloid model is a model of hyperbolic space. What's the ...

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