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

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2
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1answer
68 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 e^{-t/4}\int_{d(x,y)}^{\infty}\frac{re^{-r^2/4t}}{\sqrt{\...
1
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1answer
45 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 ...
1
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1answer
52 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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1answer
48 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 ...
3
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1answer
33 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 line-...
3
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2answers
47 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 ...
0
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1answer
77 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, ...
3
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0answers
52 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) & \cos(\theta)\...
3
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1answer
75 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 ...
2
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2answers
152 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 ...
8
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1answer
140 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
210 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 ...
4
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1answer
79 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 ...
2
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0answers
43 views

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 (...
2
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1answer
72 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 ...
2
votes
1answer
216 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 ...
0
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1answer
35 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 ...
5
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2answers
230 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 ...
3
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3answers
136 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 ...
3
votes
2answers
115 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 = \int_{t_1}^{t_2}...
2
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1answer
21 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
92 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 ...
2
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0answers
79 views

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 ...
3
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2answers
71 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 ...
2
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2answers
234 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 ...
4
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2answers
109 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 ...
2
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1answer
55 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 ...
8
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1answer
146 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 ...
0
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0answers
263 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
65 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, ...
0
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2answers
108 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 $P\...
4
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1answer
396 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 https:...
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1answer
35 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
76 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 ...
5
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2answers
94 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 $\color{red}{\text{P}}$,$\...
2
votes
2answers
130 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 ...
0
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2answers
154 views

Hyperbolic Ideal Triangle

I have everything pretty much figured out everything but I need help proving the unique point formed by the three perpendiculars in the picture
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1answer
41 views

Jacobian for Partial Iwasawa Coordinates

I am working through Terras' Harmonic Analysis, V2, and am stuck on I believe a notational point. We are asked to show that for $$Y=\begin{pmatrix}V&0\\0&W\end{pmatrix}\begin{bmatrix}I_p&0\...
0
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1answer
83 views

circumscribe a regular polygon around a circle in hyperbolic geometry [duplicate]

In the hyperbolic plane, let a circle of radius r be given. If we want to circumscribe a regular polygon with n sides around this circle (i.e., if we want the sides of the polygon to be tangents of ...
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2answers
187 views

For what $n$ does a hyperbolic regular $n$-gon exist around a circle?

Does there exist a relationship in terms of $r$ and $n$ to represent how large $n$ must be if $r$ of the circle is given in the hyperbolic plane? (The edges of the regular $n$-gon are tangent to the ...
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1answer
27 views

Hyperbolic quadrilaterals : Opposite sides of the quadrilateral cannot intersect

Suppose that a hyperbolic quadrilateral $ABCD$ satisfies $h(A, B) = h(C, D), h(B, C) = h(A, D)$. Mark each of the following claims about the quadrilateral as true or false: Opposite angles of the ...
0
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1answer
60 views

Euclidean circle in complex plane

I am reading Anderson's Hyperbolic Geometry and am having trouble with one of the Exercises in Chapter 1: Consider the unit circle $\mathbb{S}^1=\{z \in \mathbb{C} \text{ s.t. }|z|=1\}$. Let $A$ be a ...
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1answer
167 views

Circles inscribed in regular polygons in hyperbolic geometry

Does the radius of a circle matter when determining the number sides of a regular polygon in hyperbolic geometry? The sides must be tangent to the circle. Can't I just use an equilateral triangle ...
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1answer
28 views

Hyperbolloid Model Translations

Although the hyperboloid model of hyperbolic geometry has natural analogues of reflections and rotations, I am having trouble finding any linear transformation which is distance preserving and ...
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1answer
37 views

Given a non-ideal hyperbolic triangle and the Euclidean comparison triangle with equal side lengths, are the interiors of the two bi-Lipschitz?

Fix three finite real numbers $p,q,r > 0$. Up to isometry, there is a unique 2-simplex $\Delta$ in the Euclidean plane bounded by a geodesic triangle with these three reals $p,q,r$ as side-lengths. ...
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1answer
41 views

Prove this equality about hyperbolic right triangles

If K is the area of a hyperbolic right triangle ABC in which the right angle is at C, prove that $$ \sin K=\frac{ \sinh a \sinh b}{1+\cosh a\cosh b}$$ My attempt at the solution: I basically need ...
3
votes
1answer
88 views

Finding the hyperbolic length in a hyperbolic right triangle

Question: In a hyperbolic right angled triangle, the two legs have hyperbolic lengths of $3$ and $4$. What is the hyperbolic length of the hypotenuse? Is this larger or smaller than $5$? I'm having ...
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2answers
87 views

Hyperbolic Geometry and Circles

How does the angle of parallelism relate to the arc of a circle and a point outside? In Hyperbolic Geometry, I'm trying to figure out what happens to the "visibility" of a circle when a point ...
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0answers
41 views

Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $. at the ...
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3answers
216 views

Why do lines in the poincare model meet the infinite edge at right angles?

I know the lines are generated by projecting geodesics on a hyperboloid to a plane and the boundary of the disk comes from the asymptotic cone around the hyperboloid, but I just don't see why the ...