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

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Hyperbolic line with isometry. [on hold]

I have been doing some research to help solve for models of hyperbolic geometry. For a function f(z) in the following conditions are equivalent: f sends two points of l(0) to points in l(0), f is ...
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1answer
44 views

Triangular Area on hyperbolic surface

I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof ...
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26 views

Curvature of hyperbolic surface

So from what I understand the curvature of a surface by calculated by inversing the radius of the osculating circle. But if a hyperbolic surface have a negative curvature, wouldn't that imply the ...
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2answers
34 views

N-polygons in hyperbolic geometry

Let $N$ be an integer and we have two $N$-polygon $A_{1}A_{2}\ldots A_{N}$ and $A'_{1}A'_{2}\ldots A'_{N}$ such that the length of geodesic $A_{i}A_{i+1}$ is equal to the length of geodesic ...
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1answer
14 views

rotations and SU(1,1)

I'm interested in the isometries of the hyperbolic plane, i.e. the mappings which leave the geometric properties of objects invariant. I'm working with the Poincare disc model. My lecture notes ...
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42 views

Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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1answer
14 views

What's the right way to calculate hyperbolic distance on the hyperboloid model?

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model: Let $u = ...
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2answers
35 views

Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
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24 views

Hyperbolic distance with natural log

We have been learning a lot about hyperbolic lines, midpoints and distances. Would someone be able to assist me how to solve for this problem? The hyperbolic line consists of positive real numbers ...
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20 views

Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
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2answers
24 views

What is the largest possible sum of all the angle measures of a $\Delta$ in hyperbolic space?

$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?
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1answer
27 views

Hyperbolic segment from $(0,0)$ to $(0,0)$

Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work: 1) The segment must apply to the rules of ...
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1answer
35 views

Heat kernel formula on hyperbolic plane well defined

Consider the heat kernel for the hyperbolic plane $\mathbb{H}^2$ and the corresponding heat kernel: $$k(x,y,t)=\frac{C}{t^{\frac{3}{2}}}\cdot ...
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1answer
21 views

Are bounded geodesics in the modular surface closed?

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any ...
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1answer
17 views

Finite hyperbolic geometry with ideal points

I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 ...
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18 views

Equation of line in hyperbolic space

After a slightly peculiar dream the other night, I find myself suddenly inspired to do numerical simulations in three-dimensional hyperbolic space. For this to work, I need an equation of line in ...
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1answer
20 views

“Square,” line-preserving models of the hyperbolic plane

The Klein model of the hyperbolic plane is a line-preserving map from $H^2$ to the disk. Is there a model of the hyperbolic plane which is a line-preserving map from $H^2$ to $[0,1]^2$? By ...
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2answers
45 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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1answer
64 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
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0answers
38 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
2
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1answer
56 views

Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I've been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional ...
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2answers
73 views

Parallel transport of a vector in hyperbolic space, specifically in $\mathbb{H}$

Let us consider Poincaré's upper plane which is defined as $\mathbb{H} = \{ (x,y) | y>0\}$. This space has a Riemannian metric $g = \text{diag}(1/y^2, 1/y^2)$. Now let us consider a differential ...
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2answers
76 views

Totally geodesic hypersurface in compact hyperbolic manifold

In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only ...
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1answer
34 views

Mapping the Poincare disk model to the Poincare half plane model

I am puzzeling with the following: Given a point $ A = ( a_x, a_y) : a_x^2+ a_y^2 \le 1 $ in the Poincare Disk model ( https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model ) to which point does ...
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1answer
22 views

A lift of isometry to universal covering

Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than ...
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Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
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1answer
30 views

Edge length of hyperbolic tesselations

If I have a general uniform tesselation in hyperbolic plane (same configuration of regular polygons at every vertex, but multiple types of polygons allowed), how can I find the edge length and/or ...
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19 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
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1answer
38 views

Hyperbolic geometry and the Triangle Inequality

In Is the shortest path in flat hyperbolic space straight relative to Euclidean space? I answered by refering to the Triangle Inequality (https://en.wikipedia.org/wiki/Triangle_inequality , Euclid's ...
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1answer
29 views

Relationship between Hyperboloid model of hyperbolic space and disc model / confused by a picture.

I am confused by this picture: https://en.wikipedia.org/wiki/File:HyperboloidProjection.png What is wrong with projecting from the origin, and using the disc at $t = 1$? After doing some computation ...
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2answers
76 views

Classification of conics in hyperbolic plane

How many different types of conics exist in hyperbolic plane? Euclidean geometry has three, of course. But when I was trying to find out results for the hyperbolic plane, the best thing I found ...
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3answers
48 views

how to construct a line on a poincare disk?

Given the Euclidean coordinates of two points (p1, p2) and (q1, q2) in the unit circle, how do I construct the Euclidean circle x^2 + y^2 + fx + gy+1 representing the hyperbolic d-line on the poincare ...
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2answers
63 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
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1answer
19 views

Whether the hyperbolic trigonometry rule holds for all models

I am reading a paper concerning hyperbolic geometry. It represents some results like the hyperbolic cosine rule. Consider a hyperbolic triangle with side lengths $a$, $b$, $c$ and angles $\alpha$, ...
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2answers
43 views

A triangle in the Poincare disc model

Suppose that we have a triangle in the Poincare disc model such that the internal angels are all equal. Then Does it imply that the lengths of sides are all equal? By length of a side, I mean the ...
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0answers
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constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
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2answers
44 views

Travel across the Poincaré disk model of the hyperbolic plane

A is a point on the Poincaré disk model of the hyperbolic plane. B is a second point, d hyperbolic distance away from A. The hyperbolic ray AB passes through A at angle θ. How might one find the ...
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2answers
70 views

Proving pseudo-hyperbolic distance is distance

The pseudo-hyperbolic distance on the unit disk is defined as: $$\rho(z,w)=\left|\dfrac{z-w}{1-\bar wz}\right|.$$ I'd like to prove it's a distance. The real problem is, as always, the triangle ...
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2answers
90 views

Translating a Euclidean proof to hyperbolic language..

User HyperLuminal asked for help to prove the following statement: Connecting the feet of the altitudes of a given triangle, we obtain another triangle for with the altitudes of the original ...
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1answer
32 views

Question about two definitions of Teichmuller space for a surface of genus $g$

There are many equivalent definitions of Teichmuller space for a surface of genus $g\ge 2$. One of them concerns the complex structure: the Teichmuller space $\mathcal{T}(g)$ is the set of the ...
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1answer
122 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
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0answers
44 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
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1answer
24 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
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2answers
46 views

Constructions of perpendicular in hyperbolic plane

Consider the disc model of hyperbolic plane $\mathbb{D}^2$ and a line $g$ through the origin $(0,0)\in \mathbb{D}\subset\mathbb{C}$, i.e. a diameter of the circle $\partial \mathbb{D}=S^1$. Let ...
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1answer
107 views

Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
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1answer
33 views

Show that $\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $

I have been reading about "mean value theorems in number theory" such as $$\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $$ How to prove such a result? One source says it is ...
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1answer
50 views

What is the name of this (circumscribed) triangle?

I am meeting the following triangle more and more in my investigations of ideal triangles in the Beltrami Klein model of hyperbolic geometry. That made me wonder: is there a name for it? (And does it ...
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2answers
80 views

An interesting point of a triangle. (Help needed to prove a statement.)

Consider a triangle whose sides are segments of $\color{red}{\text{line}}$, $\color{blue}{\text{line}}$, $\color{green}{\text{line}}$ falling in the circum-circle $c$. Let ...
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2answers
48 views

Finding angles of hyperbolic triangles

I am trying to learn about how to find the angles of hyperbolic triangles. Now below is a problem: It has all the steps but I am not understanding the concept (the ones that are underlined in green ...
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0answers
30 views

Ideal Triangles and Klein Beltrami Disc

I'm trying to prove something with the ideal triangle in hyperbolic geometry and someone told me that the ideal triangle looks like a euclidean triangle inscribed in a circle in the Klein Beltrami ...