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

learn more… | top users | synonyms

3
votes
1answer
70 views

Various definitions of Moduli space of Riemann surfaces and Uniformization theorem

I'm sorry for the quantity of questions I'm asking, but I would like to solve once and for all many doubts I have on equivalent definitions of the moduli space of Riemann surfaces. Definition 1: The ...
1
vote
1answer
27 views

A question on Fuchsian group and automorphism of surface

Let $\mathbb{H}$ be the upper half of the complex plane, i.e., $\mathbb{H} =\{ z \in \mathbb{C}: \operatorname{Im} z >0\}$. And let $V$ be a Riemann surface with genus $g \ge 2$. Then $V$ is ...
3
votes
1answer
80 views

Area of hyperbolic triangle definition

I found this question recently in my booklet on hyperbolic geometry asking a very simple question but I could not answer it: Why can we not define the area of a hyperbolic triangle as in the plane ...
1
vote
1answer
31 views

How is $PSL(2, \mathbb{R})$ explicitly identified with the unit tangent bundle $T^1(\mathbb{H})$?

Let's say I have a given matrix $\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$, what point and tangent direction is it identified with?
2
votes
1answer
17 views

What is the relationship between the Liouvelle measure on $T^1(\mathbb{H})$, and the Haar measure $\mu$ on $PSL(2, \mathbb{R})$?

Let $\frac{dx dx d\theta}{y^2}$ be the Liouvelle measure on $T^1(\mathbb{H}) \cong \mathbb{H} \times \mathbb{S}^1$. Let $\mu$ be the Haar measure on $PSL(2, \mathbb{R})$. What is the relationship ...
3
votes
0answers
49 views

What are some good graphics programs for depicting hyperbolic geodesics?

I'm looking for some software that allows one to draw accurate pictures in hyperbolic space. In particular, I want to be able to specify pairs of points and generate the geodesic between them, in ...
1
vote
0answers
57 views

Difference between Euclidean and Hyperbolic lengths.

What is the difference between Euclidean and Hyperbolic lengths? For instance if i were to measure a curve on with the euclidean distance and alternatively the Hyperbolic distance, What would be the ...
1
vote
0answers
30 views

What is the metric of the Riemann surface resulting from quotiening the upper half plane by a Fuchsian group?

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. What is the metric we get on $\mathbb{H} / \Gamma$?
0
votes
0answers
256 views

What is the volume of sphere in hyperbolic space?

I'm looking for a formula to describe surface and volume of a sphere in hyperbolic 3-space. I found some results which were generalized for any dimension, but I wasn't able to understand them. ...
2
votes
1answer
173 views

Arc-length Parametrization of Geodesic in Hyperbolic Plane

Here's a related question. So, we have $H=\{(x,y)\in\mathbb{R}^2: y>0\}$ and define the metric $g=\frac{1}{y^2}(dx^2+dy^2)$. I know that the circle $x^2+y^2=1, y>0$ is the image of a geodesic in ...
3
votes
1answer
124 views

For this hyperbolic punctured torus, how does the Dirichlet domain change as we move its center?

Consider the Fuchsian group $\Gamma:=\Big\langle\begin{pmatrix}1&1\\1&2\end{pmatrix}, \begin{pmatrix}1&-1\\-1&2\end{pmatrix}\Big\rangle$. A commonly studied fact is that when this acts ...
0
votes
0answers
28 views

Examples of Supergroups: U(n | m), SU(N, n|m) and PSU(N, n|m).

Looking for explicit forms of group elements in the supergroups: (1) U(n | m), (2) SU(N, n|m), (3) PSU(N, n|m), (4) PSL(n|m), (5) OSp(n|m). We can simply take $N=2$, $n=1$ and $m=1$. Partial ...
1
vote
2answers
44 views

Show a set is open in C

Show that the set O={z$\in\Bbb C$:Re(z)<0, Im(z)>0} is open in $\mathbb C$ I know this is the second quadrant, but I don't know how to do the proof.
0
votes
2answers
27 views

When $\cosh yx/2=\pm 1 $?

When $\cosh \frac{xy}{2}=\pm 1 $? is it correct to say $xy/2=cosh^{-1}(\pm1)$ Then $xy=2 \cosh^{-1}(\pm1)$ I think there is better solution for this problem? any idea?
1
vote
0answers
33 views

Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
0
votes
1answer
45 views

Poincaré disk model of hyperbolic plane

Can someone please explain trigonometry in Equations (2) to (8) of: PoincareDisk ?
1
vote
1answer
44 views

What are the most important issues to consider in upper-half plane model?

I hope you can help me, I need to do a research project about this model of hyperbolic geometry. Honestly, I've never studied the subject, and I'm not sure that subjects should give more importance. ...
4
votes
1answer
70 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
7
votes
2answers
74 views

Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
4
votes
3answers
130 views

Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
8
votes
2answers
373 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
0
votes
0answers
150 views

Isometries of hyperbolic space

The metric tensor for the Poincaré ball model of hyperbolic geometry is $$ g_{ij} = \frac{\delta_{ij}}{(1 - \lvert \mathbf{r} \rvert^2)^2} $$ where $\mathbf{r}$ is the position in the ambient ...
0
votes
0answers
47 views

Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
2
votes
0answers
111 views

Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
2
votes
1answer
25 views

Why to include the $C$ in the formula for the distance in hyperbolic geometry?

I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would ...
1
vote
0answers
26 views

Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
-1
votes
1answer
81 views

How (not) to plot a Hyperbola? [closed]

I am trying to plot a Hyperbola in Wolfram Alpha. Its giving me a strange graph. How to correct that?
4
votes
2answers
72 views

Error when computing geodesics in hyperbolic half plane

It is known that the geodesic equations for the upper half plane equipped with the hyperbolic metric are $$x''=\frac{2x'y'}{y},$$ $$y''=\frac{(y')^2 -(x')^2}{y}.$$ It is also well known that the ...
3
votes
1answer
54 views

Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) \...
0
votes
1answer
108 views

Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through ...
4
votes
1answer
309 views

Poincaré hyperbolic geodesics in half-plane and disc models

The objective of this post is to state that 1) the Poincaré hyperbolic metric results in a solution of complete geodesic circles in both half-plane and disk models. 2) the choice of one or other ...
0
votes
0answers
14 views

Cross Ratio of two rays through origin

There are two fixed and two variable concurrent rays of unit length in 3 space through the origin. How should the spherical coordinates of the two variable points be related to result in a constant ...
0
votes
1answer
33 views

The shape of the hyperbolic curves coordinates

Any one has an idea about hyperbolic coordinates ? and how to imagine it ? Indeed I am trying to find the shape of the coordinate curves far away from origin ! and what is the shape of them at $u=0 ,...
2
votes
3answers
224 views

Distance formula for points in the Poincare half plane model on a “vertical geodesic”.

In comment at http://math.stackexchange.com/a/1381829/88985 at Distance of two hyperbolic lines is says (as i interpreted it) that the distance between two points $(a,r)$ and $(a, R)$ in the ...
2
votes
1answer
44 views

Equidistant points to a hyperbolic line

consider the Poincare upper half-plane model of hyperbolic plane $\mathbb{H}^2$ and a hyperbolic line $\ell\subset \mathbb{H}^2$ (or geodesic if you want). I would like to visualize the set of points ...
1
vote
1answer
44 views

Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
1
vote
1answer
118 views

Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
1
vote
1answer
54 views

On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?

Wikipedia has the answer in the case of the Poincaré disk model. When the point $\mathbf{v}$ is translated to the origin, then the point $\mathbf{x}$ is translated to $$\frac{(1 + 2\mathbf{v} \cdot \...
3
votes
2answers
165 views

Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, \ell_2:=\...
1
vote
1answer
101 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 ...
0
votes
0answers
72 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 ...
3
votes
2answers
45 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 $A'_{i}A'...
0
votes
1answer
27 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 ...
1
vote
0answers
65 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 $\{3,7\}$), ...
0
votes
1answer
34 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 = (...
1
vote
2answers
62 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 ...
2
votes
0answers
35 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 ...
0
votes
2answers
36 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 $m\angle A+m\angle B+m\angle C$?
0
votes
1answer
32 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 ...
2
votes
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{\...