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

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Geodesics in Poincare Disk

I would like to find the geodesics in the Poincare disk. I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using ...
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uniquess of hyperbolic numbers

I'm trying to prove, the uniqueness of hyperbolic numbers, like the complex numbers are unique, but since hyperbolic numbers aren't a field, I can't use the ideas of this. Are there theorems of ...
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How to parameterize these pretty hyperbolic surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
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+50

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
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1answer
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This map is an isometry (in the Riemannian sense) of the hyperbolic plane. Why is the following a proof of it?

I'm making my way through a textbook on elementary undergraduate geometry. The author has defined the notion of an isometry between two subsets of $\mathbb{R}^2$ equipped with a Riemannian metric. It ...
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1answer
25 views

Examples of Lattices in $\operatorname{Isom}(H^n)$ for all $n \geq 2$?

I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples. ...
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1answer
21 views

Triangle inequality for hyperbolic metric of logarithm of cross ratio.

Consider the Poincaré Half-Plane model of the Hyperbolic Space $ \mathbb{H}^2 $. I need to proof that the following d function is a metric.$ d:\mathbb{H}^2\rightarrow\mathbb{R}, d(A,B) = \big{|} log ...
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1answer
34 views

Length of the arc of a hypercycle

I am still puzzeling to get a nice equation for the arclength of an hypercycle. (I asked a similar question (less developed) about a year ago that was never answered, now i am a bit further, i ...
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2answers
48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
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problem with proof 'horoball QI extension theorem'

I'm reading the book of Drutu and Kapovich "Lectures on geometric group theory". In the proof of Mostow rigidity theorem, they say that they can extend an $\rho$-equivariant function ...
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Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
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Need to change variables in equations with cosh.

i have these five functions: $x=\tau \cosh(s)$ $q=\tau \sinh(s)$ $y= \sinh(s)$ $p= \cosh(s)$ $u= 1/2*\tau*\cosh(2s)+1/2*\tau$ I need to write $u$ in terms of $x$ and $y$ I know the answer is ...
2
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1answer
29 views

Non-equivalent metrics on $PSL_2(\mathbb{R})$

I am reading a paper on continued fractions and it uses the following result on Lie Groups: Fix an arbitrary left-invariant metric $d$ on $PSL_2(\mathbb{R})$ ... This phrase really throws me ...
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1answer
51 views

How to analytically derive the geometric properties of a hyperbola and programatically use them to graph?

I have the following equation: $$ \left(10^{\left\lfloor\frac{\ln\sqrt k}{\ln10}\right\rfloor-1}+x\right) \left(10^{\left\lfloor\frac{\ln\sqrt k}{\ln10}\right\rfloor-1}+y\right) = k $$ Entering the ...
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1answer
11 views

Finding the second point of intersection from a normal on a hyperbola

The question is phrased as follows: "A rectangular hyperbola, W, has equation xy = 12" a) Show that the gradient of the normal, N, to W at the point P(2,6), is 1/3. b) Hence find an equation for ...
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0answers
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congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
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1answer
46 views

Equalities and inequalities for quadrilaterals in hyperbolic space

In euclidean space any quadrilateral satisfies equalities and inequalities $$a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2$$ $$a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2$$ where $a,b,c,d$ are the side lenghts, ...
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Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
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Hartshorne Exercise 41.2 - Altitudes of a triangle have a common perpendicular [hyperbolic]

This is my first post here ever, so don't be too rude, if i missed something. My question refers to exercise 41.2 "GEOMETRY:EUCLID AND BEYOND" from Robin Hartshorne. You can easily find the book as ...
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1answer
31 views

Discrete group of isometries of a finitely compact metric space is countable.

This question comes from Ratcliffe's Foundations of Hyperbolic Manifolds. Let $X$ be a finitely compact metric space (i.e. all closed metric balls are compact). Prove that a discrete group ...
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Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article "inavariant metrics on ...
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Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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1answer
28 views

Relationship between regular tessellations in general space

Using the notation for regular tessellations $\{p,q\}$ denoting the tessellation consisting of p-gons , q of which meet at each vertex [e.g. $\{3,6\}$ in $\mathbb{R}^{2}$ is the equilateral triangle ...
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1answer
16 views

Incidence axioms for the upper half plane (complex plane).

The axiom I am checking for this question is I1: "For any two distinct points A,B there exist a unique line L containing both points." Show that I1 is satisfied in the upper half of the complex plane ...
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another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition ...
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2answers
58 views

What is the radius of the inscribed circle of an ideal triangle

I wanted to calculate the radius of the inscribed circle of an ideal triangle. and when i dat calculate it i came to $\ln( \sqrt {3}) \approx 0.54 $ (being arcos(sec (30^o)) but then at ...
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1answer
30 views

Where are the vertices of the universal cover of the genus 2 torus octagon?

The universal cover of the genus 2 torus is hyperbolic plane and the fundamental domain is a octagon. Here is a picture, which I took from here. Is there a closed form for the points of set of the ...
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1answer
40 views

In hyperbolic geometry, prove that parallel lines are not equidistant

In Euclidean Geometry, parallel lines are equidistant. In hyperbolic geometry, it appears that parallel lines are $not$ equidistant. Is there a proof that supports this, or is it supposed to be ...
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1answer
24 views

Hyperbolic geometry question concerning lengths between parallel lines

Theorem (H16). If: $l$ and $m$ are parallel lines, $j$ is a common perpendicular intersecting $l$ at point A and $m$ at point B, and C and E are points on $l$ so that C is between A and E, Then: ...
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1answer
70 views

Lie groups for beginners: Lie group of hyperbolic geometry

I am trying to understand Lie groups and their relation to (2 dimensional) hyperbolic geometry. as far as I understand it (which is not very far, I am pushing my understanding here) the Lie-group ...
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1answer
25 views

Conformal Map from Intersection of Two Discs and Half-Plane

I have one of those "find the map" problems that is really giving me a lot of trouble. Let $B_1(1)$ be the ball of radius $1$ centered at $1$. We have the following domain: $\mathbb{D} \cap B_1(1) ...
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2answers
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How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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1answer
22 views

Isoperimetric inequality in the Poincare disc

In the Poincare disc model, a version of the isoperimetric inequality states that $L(\partial(A))>c\mu(A),$ where $\mu$ is the hyperbolic area and $L(\gamma)=\int_0^1 ...
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spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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1answer
44 views

Volumes of hyperbolic manifolds

In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental?
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Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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0answers
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What is a cusp neighborhood corresponding to a parabolic Mobius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
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Prove $A \cdot B\leq -1$, where $A$ and $B$ are in $\mathbb{H}^2$

Let $A$ and $B$ be in $\mathbb{H}^2$. I need to prove that the lorentzian dot product between $A$ and $B$ is less than or equal to $-1$. I have no idea where to start.
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1answer
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Non euclidean lines (finding endpoints of semicircle)

A non euclidean line in $\mathbb{RP}^1$ in terms of reflections about the unit circle can be written in the form $A+B(\overline{w}+w)+C(\overline{w}w)=0$ Where $w=\frac{1}{\overline{z}}$ The ...
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22 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
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Villarceau circle as a Loxodrome

A circular Clifford torus (radius at flat circle = h, section radius $ a , a<h $ ) is cut by a plane at an angle $ \cos \alpha = a/h$ centrally to the symmetry axis, the line of intersection is a ...
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Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as ...
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1answer
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Can a hyperbolic surface be isometrically embedded into $\mathbb R^4$?

Can a complete hyperbolic surface be isometrically embedded into flat $\mathbb R^4$?
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When a ray of an horocircle passing through the origin intersects the y axis.

In the following figure, $h(A,B)$ is an horocycle centered in A passing over B. $\Theta(h)$ is the angle of parallelism of the segment $h$ and $S$ is the well known intersection of a chord of an ...
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1answer
36 views

Tiling on Poincaré disc [closed]

Is there anyone to help me tile on a Poincaré disc? In fact, I'm going to tile triangle tiles on a surface in hyperbolic geometry ; is there any algorithmic method to do so?
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Mapping the Poincaré disk to hyperbolic surfaces in $\mathbb{R}^3$.

Take any hyperbolic surface with constant curvature in $\mathbb{R}^3$, such as Dini's surface, or a hyperboloid of constant curvature. If I understood things correctly, for any such surface, we ...
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How to make a triangular pillow with a pre-drilled tube with two truncated tetrahedra? (from Jeff Weeks' paper)

In his paper Computation of Hyperbolic Structures in Knot Theory, p.12, Jeff Weeks explains as below how to make a triangular pillow with a pre-drilled tube by gluing two truncated tetrahedra. A ...
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1answer
57 views

Compute of curvature

In the answer of this question,for the given metric $$g_M = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2,$$ how to compute the curvature? Whether the hyperbolic space means $M=\{x\in R^n:x_n>0\}$? ...